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ELEMENTARY  ALGEBRA 


BY 
GEORGE  W.  MYERS 

THE   UNIVERSITY   OF   CHICAGO 

AND 

GEORGE  E.  ATWOOD 

NEWBURGH,    NEW   YORK 


if 


SCOTT,  FORESMAN  AND  COMPANY 
CHICAGO  NEW  YORK 


Ml 


COPYRIGHT  1916 

BY 

SCOTT,  FORESMAN  AND  COMPANY 

cLUUCm  i  ION   OEf'L 


PREFACE 

The  authors  make  no  apology  for  offering  another  algebra 
to  the  school  public.  In  influential  places  algebra  has  been 
challenged  as  a  suitable  subject  for  high  school  pupils.  Is 
it  not  the  part  of  wisdom,  before  eliminating  a  subject  of  so 
long  and  undisputed  standing  as  algebra,  to  try  reconstruct- 
ing and  improving  its  form  and  even  some  of  its  substance? 
The  authors  believe  that  this  text  has  accomplished  much 
in  both  of  these  particulars. 

This  book  is  not  written,  however,  with  the  thought  of 
defending  an  unworthy  claimant  to  a  place  in  the  curriculum. 
The  true  view  is  that  the  high  educational  merit  of  school 
algebra  may  be  raised  even  higher  by  a  treatment  whose 
language  and  mode  of  exposition  are  in  accord  with  the  possi- 
bilities and  appreciations  of  youth,  and  whose  scientific 
soundness  is  at  the  same  time  not  seriously  compromised. 
It  is  the  authors'  conviction  that  rightly  taught,  algebra  is  of 
great  educational  value,  and  that  to  most  high  school  students 
it  is  not  distasteful. 

In  carrying  out  their  views  on  this  line,  the  authors  have 
attempted  several  specific  things.  Some  of  these  stated 
briefly  are  as  follows: 

1.  To  present  the  material  in  a  language  and  mode  that  are 
simple  and  at  the  same  time  mathematically  sound,  without 
resort  to  mathematical  technicalities. 

2.  To  motivate  the  various  topics  of  algebra  either  through 
special  problematic  situations,  or  through  the  gradually 
rising  demands  of  the  equation  for  particular  phases  of  alge- 
braic technique.     As  examples  see  pages  27,  32,  59,  266,  etc. 

3.  Persistently  to  make  the  first  steps  into  the  treatments 
of  algebraic  subjects  through  the  analogous  subjects  of  arith- 
metic.    (See  pages  20,  41,  91,  107,  180,  229,  etc.) 

__  iii 

54! 29  J 


iv  PREFACE 

4.  To  give  the  pupil  some  really  valuable  help  in  learning 
to  read,  to  comprehend,  and  to  interpret  algebraic  language, 
and  to  express  mathematic  principles  and  rules  in  this  lan- 
guage. Chapter  XIII  on  General  Numbers,  Formulas,  and 
Type-forms  may  be  cited  as  a  good  illustration  of  this 
treatment. 

5.  To  give  an  early  introduction  to  simultaneous  simple 
equations  and  to  complete  their  study  by  recurrent  treatments 
as  the  course  develops. 

6.  To  make  early  and  frequent  use  of  the  graph  freed  from 
analytical  technicalities,  as  an  aid  to  the  development  of  alge- 
bra through  clarifying  and  vivifying  meanings  of  algebraic 
processes  and  technique  in  the  beginning  stages  of  teaching 
and  learning  them. 

7.  To  seek  diligently  for  such  an  order  of  treatment  of  the 
special  topics  as  is  dictated  by  the  highest  economy  in  the 
mastery  of  the  elements  of  the  science  of  algebra.  By  this 
means  it  is  hoped  to  give  a  stronger  and  a  more  highly  edu- 
cative first-year  course  in  the  customary  time.  (See  Table 
of  Contents.) 

8.  Carefully  to  grade  as  to  difficulty  and  to  balance  as  to 
quaUty  and  quantity  the  problems  and  exercises  of  the  book, 
again  with  an  eye  single  to  the  unfolding  needs  of  algebra. 
(See  problem-lists  given  under  the  different  topics.) 

9.  To  correlate  with  arithmetic,  geometry,  general  science, 
and  everyday  life  to  as  great  a  degree  as  the  best  school 
interests  of  first-year  algebra  require. 

10.  To  heighten  the  workability  of  the  text  by  a  synoptic 
table  of  contents,  a  summary  of  definitions  (page  322),  and 
a  good  working  index. 

A  little'  of  the  pedagogical  background  of  the  organiza- 
tion of  this  text  may  be  stated  here.  Tiie  authors  hold  the 
view  that  teachers  of  present-day  secondary  algebra  should 
recognize  that  they  are  under  three  significant  professional 
obligations  to  their  pupils,  viz. : 


PREFACE  V 

I.  To  rationalize  the  analogous  arithmetic  of  the  algebraic 
topics  taught. 

It  is  hardly  reasonable  to  expect  of  beginners  in  secondary 
algebra  that  they  really  understand  their  arithmetic,  even 
as  arithmetic.  Still  less  may  secondary  teachers  rightfully 
expect  that  beginning  pupils  have  grasped  their  arithmetic 
in  such  form  that  it  can  be  made  the  basis  for  algebra.  This 
is  a  much  more  difficult  matter  because,  although  both  arith- 
metic and  algebra  are  abstract  sciences,  algebra  involves  a 
much  higher  order  of  abstractness  than  arithmetic. 

In  view  of  the  scope  and  complexity  of  modern  elementary 
school  arithmetic,  of  the  sUght  emphasis  of  school  officials, 
examiners,  and  surveyors,  and  even  of  school  programs  upon 
rationalizing  processes,  it  is  worse  than  useless  to  expect,  let 
the  most  conscientious  teacher  strive  as  he  may,  that  more  be 
done  in  the  elementary  school  than  to  rationalize  the  most 
elementary  notions  and  processes  of  arithmetic.  In  fact,  for 
several  years  elementary  teachers  have  been  urged  by  some 
authorities  to  renounce  rationalizing  for  mere  habituating  and 
drill  procedures.  These  things,  coupled  with  the  fact  that 
arithmetic  of  the  sort  covered  in  our  grammar  grades  is  one 
of  the  most  difficult  of  all  mathematical  branches,  and  with 
limitations  of  program  time  and  immaturity  of  pupils,  hope- 
lessly preclude  any  attempts  at  those  far-reaching  inductions 
and  generalizations  that  are  essential  at  the  very  beginning 
of  rational  algebra.  Therefore,  this  fundamental  work  for 
the  highly  specialized  needs  of  the  several  algebraic  topics 
belongs  properly  to  the  algebra  teacher.  This  text  supplies 
the  initiatory  arithmetical  rationalizing  for  the  algebraic 
topics  and  subjects  at  the  precise  places  where  it  is  needed 
and  of  the  sort  that  is  appropriate. 

II.  To  show  that  many  algebraic  things  can  be  done  geomet- 
rically, i.e.,  by  the  aid  of  the  concrete  space  material  of 
diagrams,  pictures, — of  any  graphical  helps  to  clear  thinking. 

To  see,  to  calculate,  and  to  comprehend  is  the  true  order  of 


vi  PREFACE 

steps  in  mastering  algebraic  tasks.  The  concepts  of  lines, 
rectilinear  figures,  and  solids  are  so  much  space  material, 
always  and  everywhere  available  for  concreting,  visualizing, 
and  vivifying  number  laws  and  relations,  at  no  great  cost 
in  money  or  effort.  The  high  school  youth  has  lived  long 
enough  in  this  world  of  space  to  have  become  familiar  with  it, 
and  his  spatial  experiences  need  only  to  be  drawn  upon  to 
enable  him  to  lay  firm  hold  on  the  highly  abstract  fundamen- 
tals of  beginning  algebra.  Really  to  see  that  algebra  merely 
generalizes  mensuration  laws,  that  algebraic  numbers,  laws, 
and  problems  picture  into  vivid  forms,  and  to  learn  the  secret 
of  laying  before  his  eyes  diagrammatically  the  conditions  of 
algebraic  problems  as  an  aid  in  formulating  these  conditions 
into  algebraic  language  and  technique,  are  of  the  highest 
interest  and  value  to  the  beginner.  The  professional  duty 
of  employing  the  concreting  agencies  of  pictures,  diagrams, 
geometrical  figures,  and  graphs  to  vivify  and  vitalize  algebra 
will  be  readily  accepted  by  the  teacher  who  strives  to  realize 
in  practice  the  educational  merits  of  well-taught  algebra. 
No  clumsy  laboratory  equipment  of  extensive  and  expensive 
apparatus  is  required  to  enable  the  algebra  teacher  through 
space-materials  to  supply  genetic  backgrounds  for  algebraic 
problems,  truths,  and  laws. 

III.  To  show  the  pupil  that  algebra  will  enable  him  to  do  much 
more  than  he  can  do  with  either  arithmetic  or  geometry,  or  both.  , 

The  first  and  second  professional  duties  are  really  prelimi- 
nary, through  which  motivating  and  clearing  the  way  for 
effective  attack  are  accomplished.  This  third  duty  is  pecu- 
liarly due  to  algebra.  It  is  in  fact  due  to  both  pupil  and  sub- 
ject that  the  particular  gains  to  be  secured  by  a  mastery  of  the 
subject-matter  shall  appear  in  the  learning  acts. 

For  example,  the  pupil  should  see  such  things  as,  that  by 
arithmetic  he  cannot  subtract  if  the  subtrahend  happens  to  be 
greater  than  the  minuend;  that  he  cannot  solve  so  simple 
an  equation  as  x+9  =  3;  but  that  if  he  include  the  negative 


PREFACE  vii 

numbers  among  his  number  notions  he  can  do  both  easily. 
He  should  see  that  he  can  square  and  cube  numbers  geo- 
metrically, but  that  he  can  go  no  further  with  involution  than 
this.  If,  however,  he  will  learn  the  symbohsm  of  algebra  he 
may  easily  express  and  work  with  4th,  5th,  6th,  even  with  nth 
powers.  He  should  be  shown  that  while  he  can  solve  equa- 
tions in  one,  two,  and  perhaps  in  three  unknowns  with 
graphical  pictures,  i.e.,  geometrically,  the  great  power  he  gains 
by  mastering  the  algebraic  way  enables  him  to  go  right  on 
easily  to  the  solution  of  simultaneous  equations  in  4,  5,  6, 
and  even  n  unknowns.  He  should  be  made  to  feel  that  while 
arithmetic  would  enable  him,  by  a  slow  process  of  feeling 
about,  to  find  one  solution  of  many  problems,  algebra,  if  he 
will  learn  its  language  and  method,  will  lead  him  directly  not 
to  one,  but  to  all  possible  solutions.  It  will  thus  enable  him 
to  know  when  he  has  solved  his  problem  completely.  These 
and  similar  gains  of  power  over  quantitative  problems  are  the 
real  reasons  why  the  educated  man  of  today  cannot  afford 
not  to  know  algebra.  Let  teachers  perform  this  professional 
duty  well  and  the  foes  of  algebra  as  a  school  subject  will  be 
confined  to  those  who  are  ignorant  of  it.  The  one  who  has 
learned  the  subject  will  then  regard  it  as  the  emancipator 
of  quantitative  thinking. 

It  is  desired  to  call  particular  attention  to  the  introductory 
pages  on  Reasons  for  Studying  Algebra,  and  to  Suggestions  on 
Problem-solving  on  page  113,  and  to  the  careful  treatment  of 
factoring.  The  treatment  of  the  function  notion,  on  pages 
50-56,  will  appeal  to  many  teachers.  It  will  be  noted  also 
that  this  elementary  course  is  divided  into  half-year  units. 

The  problem  and  exercise  lists  are  full,  varied,  and  carefully 
chosen.  Teachers  who  employ  supplementary  lists  of  exer- 
cises with  the  regular  text  should  not  require  pupils  to  try  to 
solve  all  the  problems  and  exercises  given  here.  These  hsts 
are  made  full  and  varied  to  afford  choice  and  range  of  material. 
Great  care  has  been  exercised  to  cover  all  the  standard  diffi- 


viii  PREFACE 

culties  of  first-year  algebra,  for  this  book  makes  its  primal 
task  to  teach  good  algebra. 

This  text  is  to  be  followed  presently  by  a  second  course  on 
Intermediate  Algebra.  The  two  together  will  cover  the 
standard  requirements  of  secondary  algebra. 

The  pleasant  task  now  remains  to  acknowledge  the  assist- 
ance the  authors  have  received  from  Mr.  John  DeQ.  Briggs 
of  St.  Paul  Academy,  St.  Paul,  Minn. ;  from  the  Misses  Ellen 
Golden  and  Estelle  Fenno  of  Central  High  School,  Washing- 
ton, D.  C;  and  from  Professor  H.  C.  Cobb  of  Lewis  Institute, 
Chicago,  all  of  whom  read  and  criticized  the  proofs  of  the 
book.  Their  criticisms  and  suggestions  have  resulted  in 
numerous  improvements. 

May  this  book  find  friends  amongst  teachers  and  pupils, 
and  a  deserving  place  amongst  the  influences  now  making  for 
the  improvement  of  the  educational  results  of  high  school 
^^Sebra.  rp^^^  Authors. 

Chicago,  September,  1916. 


CONTENTS 

FIRST  HALF-YEAR 
chapter  page 

Introduction.     Reasons  for  Studying  Algebra 1 

I.     Notation  in  Algebra.     The  Equation 7 

Notation 7 

The  Equation 11 

Axioms 13 

Directions   for    Making    Statements    and    Solving 

Problems 16 

II.     Positive  and  Negative  Numbers.     Definitions 20 

Positive  and  Negative  Numbers. 20 

Definitions 24 

III.  Addition ■ 27 

Addition  of  Monomials 27 

Adding  Similar  Terms 28 

Adding  Dissimilar  Terms 30 

Addition  of  Polynomials 32 

IV.  Subtraction.     Symbols  of  Aggregation 35 

Subtraction  of  Monomials 35 

Subtracting  Similar  Terms 36 

Subtracting  Dissimilar  Terms 37 

Subtraction  of  Polynomials 39 

Symbols  of  Aggregation 41 

Addition  of  Terms  Partly  Similar 48 

Subtraction  of  Terms  Partly  Similar 49 

V.     Graphing  Functions.     Solving    Equations    in   One 

Unknown  Graphically 50 

Graphing  Functions 50 

Solving  Equations  in  One  Unknown  Graphically 55 

Summary 58 

VI.     Equations.     General  Review 59 

Equations 59 

Clearing  Equations  of  Fractions 66 

General  Review 70 

VII.     Graphing  Data.     Solving  Simultaneous  Equations 

Graphically 74 

Graphing  Data 74 

Solving  Simultaneous  Equations  Graphically 82 

ix 


X  CONTENTS 

CHAPTER  PAGE 

VIII.     Simultaneous    Simple    Equations.     Elimination    by 

Addition  or  Subtraction 85 

Simultaneous  Simple  Equations 85 

Elimination  by  Addition  or  Subtraction 87 

IX.     Multiplication 91 

The  Sign  of  the  Product 91 

The  Exponent  in  the  Product 93 

Multiplying  One  Monomial  by  Another 93 

Powers  of  Monomials 95 

Multiplying  a  Polynomial  by  a  Monomial 96 

Multiplying  a  Polynomial  by  a  Polynomial 97 

X.     Simple  Equations 100 

XI.     Division 107 

Dividing  a  Monomial  by  a  Monomial 107 

Dividing  a  Polynomial  by  a  Monomial .' 109 

Dividing  a  Polynomial  by  a  Polynomial 110 

XII.     Applications  of  Simple  Equations.     Elimination  by 

Substitution 113 

Suggestions  on  Problem-Solving 113 

Elimination  by  Substitution 120 

XIII.  General  Numbers.     Formulas.     Type-forms 123 

General  Numbers 123 

Formulas 124 

Forms  and  Type-forms  of  Algebraic  Numbers 130 

XIV.  Factoring 134 

Monomial  Factors  (Type-form:  ax-\-ay-\-az) 134 

Common  Compound  Factor:  {ax-{-ay-\-bx-\-by) 135 

Square  of  the  Sum  of  Two  Numbers:  {a^ -\-2ab i-b^) ...  137 

Square  of  the  Difference  of  Two  Numbers  (a^  -2ab+b'^)  1 38 

Trinomial  Squares:  {x^=i=2xy-\-y^) I40 

Product  of  the  Sum  and  Difference  of  Two  Numbers : 

(o+6)(a-6) 142 

Difference  of  Two  Squares  (a^-b^) 143 

Product  of  Two  Binomials  with  a  Common  Term : 

(x4-a)(x+&) 147 

Special  Quadratic  Trinomials:  {x^-\-ax-\-b) 148 

The  General  Quadratic  Trinomial:  (ax^+bx+c) 149 

Incomplete  Trinomial  Squares:   (x*+xV+?/^) 151 

Difference  of  the  Same  Odd  Powers:  (x^—y^) 153 

Sum  of  the  Same  Odd  Powers:   (x^-^y^) 154 

Review 156 


CONTENTS  xi 

CHAPTER  PAGE 

SECOND  HALF-YEAR 

XV.    Equations.     Exercises  for  Review  and  Practice 158 

Solution  of  Equations  by  Factoring 158 

Exercises  for  Review  and  Practice 164 

XVI.     Highest  Common  Factor.     Lowest  Common  Multiple  172 

Highest  Common  Factor 172 

Highest  Common  Factor  of  Monomials . .  172 

Highest  Common  Factor  of  Polynomials  by  Factoring  173 

Lowest  Common  Multiple 175 

Lowest  Common  Multiple  of  Monomials. 175 

Lowest  Common  Multiple  of  Polynomials  by  Fac- 
toring   176 

XVIL     Fractions 179 

Reduction  of  Improper  Fractions 184 

Reduction  of  Mixed  Expressions 186 

Lowest  Common  Denominator 187 

Addition  and  Subtraction  of  Fractions 188 

MultipHcation  of  Fractions 191 

Division  of  Fractions 193 

XVIII.     Literal   and   Fractional   Equations.     Solution   of 

Formulas 198 

Literal  and  Fractional  Equations 198 

Special  Methods 201 

General  Problems 207 

Solution  of  Formulas 210 

XIX.     Simultaneous  Simple  Equations 213 

Elimination  by  Comparison .  .  .  , 213 

Problems  in  Simultaneous  Simple  Equations 219 

Three  or  More  Unknown  Numbers 226 

XX.     Proportion.     Variation 229 

Ratio 229 

Proportion 232 

Principles  of  Proportion 235 

Variation 241 

XXI.     Powers.     Roots 244 

Involution 244 

Power  of  a  Power 245 

Power  of  a  Product 246 

Power  of  a  Fraction 246 

Powers  of  Binomials 247 


Xll 


CONTENTS 


CHAPTER  PAGE 

Powers.    Roots— Continued 

Evolution 250 

Root  of  a  Power 251 

Root  of  a  Product. 251 

Root  of  a  Fraction 252 

Number  of  Roots 252 

Imaginary  Roots 253 

Signs  of  Real  Roots 253 

To  Find  the  Real  Roots  of  Monomials 254 

Square  Root  of  a  Polynomial 254 

Square  Root  of  Numbers 259 

To  Find  the  Square  Root  of  a  Decimal 261 

To  Find  the  Square  Root  of  a  Common  Fraction . .  .   262 

XXII.  Exponents.     Radicals 263 

Exponents 263 

Radicals 264 

SimpUfication  of  Radicals 267 

To  Reduce  a  Mixed  Number  to  an  Entire  Surd 270 

Addition  and  Subtraction  of  Surds 270 

To  Reduce  Surds  to  the  Same  Order 271 

Multiplication  of  Surds 272 

Division  of  Surds 274 

Rationalizing  Surds 275 

Square  Root  of  Binomial  Surds 277 

Approximate  Values  of  Surds 278 

Irrational  Equations  in  One  Unknown 278 

XXIII.  Quadratic  Equations 282 

The  Graphical  Method  of  Solution 282 

Solving  Quadratics  by  Factoring 284 

Square  Root  Method  of  Solution 287 

To  Complete  the  Square  When  a  is  1 288 

To  Complete  the  Square  when  a  is  not  1 289 

Solution  by  Formula 291 

To  Find  Approximate  Values  of  Roots  of  Quadratic 

Equations 292 

Equations  in  Quadratic  Form 293 

Graphical  Solution  of  Quadratics 295 

Character  of  the  Roots  of  Quadratic  Equations 298 

To  Form  a  Quadratic  Equation  with  Given  Roots 300 

Factoring  by  Principles  of  Quadratics 301 

Problems  in  Quadratic  Equations 302 

XXIV.    Simultaneous  Systems  Solved  by  Quadratics  — ,  —  305 

Summary  op  Definitions 322 

Index 331 


INTRODUCTION 


REASONS  FOR  STUDYING  ALGEBRA 

The  high  school  pupil  should  become  convinced,  as  early 
as  possible,  that  there  are  strong  reasons  why  he  should 
learn  algebra.  The  kind  of  work  the  pupil  will  do  and  his 
consequent  sense  of  its  actual  value  to  him,  depend  so 
largely  on  the  approval  he  gives  to  its  study  that  it  seems 
worth  while,  even  before  beginning  it,  to  consider  the  reasons 
for  studying  algebra. 

ALL  TASKS  REGARDED  AS  PROBLEMS  TO  BE  SOLVED 

Whether  a  pupil  continues  in  school  or  leaves  early  for  the 
work  of  life,  he  will  soon  learn  that  the  best  way  to  deal  with 
the  questions  and  difficulties  that  arise,  is  to  regard  them  as 
problems  to  be  solved,  and  to  attack  them  as  such.  How 
to  learn  his  lessons,  to  write  a  composition,  to  do  an  experi- 
ment, to  debate  a  question,  to  win  in  a  contest,  to  do  any- 
thing the  first  few  times,  are  famiUar  problems  to  the  high 
school  pupil. 

How  to  earn  more  and  waste  less,  to  manage  affairs  more 
economically,  to  get  more  out  of  and  to  put  more  into  life, 
how  to  conduct  household  affairs  more  economically,  to 
learn  to  appreciate  and  to  understand  more  of  the  really 
good  and  true  in  books  and  in  life,  are  actual  problems  to 
every  right-minded  man  and  woman.  Right  living  is  little 
more  than  solving  a  continuous  chain  of  problems.  The 
question  for  every  young  person  should  be,  ''How  far  can 
I  advance  in  the  problem-book  of  the  great  world  before  the 
problems  get  too  hard  for  me?" 


2  ELEMENTARY   ALGEBRA 

IMPORTANT  TO  ACQUIRE  POWER  AND  SKILL  IN  PROBLEM  WORK 

Clearly  then,  it  is  of  great  importance  to  learn  what  it 
means  to  «olve  a  problem  and  to  acquire  whatever  skill  we 
may  in  the  art  of  problem-solving,  and  this  too,  not  merely 
because  our  teacher,  or  our  parents  want  us  to  do  so,  but 
for  our  own  sakes  purely.  In  an  especial  sense  algebra 
teaches  the  tactics  and  the  technique  of  problem-solving. 
The  tools  by  which  both  the  science  and  the  art  are  wrought 
out  are  the  algebraic  number  and  the  algebraic  equation. 
To  be  without  the  ability  to  use  the  equation  skillfully  is  to 
be  without  the  ability  to  do  much  problem-thinking.  Power 
to  use  the  equation  with  skill  and  insight  is  the  main  part 
of  the  equipment  of  an  accurate  thinker,  and  algebra  is 
essentially  the  science  and  the  art  of  the  equation. 

TWO  REASONS  WHY  ALGEBRA  SHOULD  BE  STUDIED  BY  ALL 

Every  person  who  has  his  way  to  make  in  the  world  must 
succeed  or  fail  in  his  struggle  with  life's  problems.  The 
world's  problems  are  harder  than  those  of  algebra,  but  the 
best  way  to  acquire  ability  to  grapple  with  harder  problems 
is  first  to  get  some  skill  with  easier  ones.  Algebra  starts 
with  comparatively  simple  difficulties  that  gradually  increase 
in  complexity  as  one's  skill  grows,  to  difficulties  great 
enough  to  tax  the  powers  of  even  the  brightest  pupils. 

For  two  reasons  at  least,  the  problem-solving  of  algebra 
is  easier  than  that  of  everyday  life. 

In  the  first  place,  the  language  of  algebra  makes  reasoning 
easier  than  does  any  other  language  men  have  yet 
devised. 

Before  algebraic  language  was  invented,  the  ancient 
mathematicians  used  ordinary  words  and  sentences  in  the 
problems  they  attempted.  The  form  of  their  work,  which 
was  largely  sentence-making,  is  now  called  rhetorical  algebra. 
It  never  amounted  to  much  as  a  problem-solving  instrument. 


REASONS   FOR  STUDYING  ALGEBRA  3 

Mathematicians  later  made  use  of  abbreviated  words, 
phrases,  and  even  sentences  that  occurred  frequently  in 
problems,  of  initial  letters,  suggestive  symbols,  and  thus 
formed  what  is  now  called  abbreviational,  or  syncopated 
algebra.  This  was  a  real  advance,  and  a  very  fair  sort  of 
algebra  now  developed  as  the  need  for  it  came  along,  and 
men  grew  interested  in  it.  But  it  was  still  cumbersome, 
and  men  continued  trying  to  improve  it  in  this  way  and 
that,  until  finally  after  many  centuries,  they  hit  upon  the 
modern  form  of  writing  algebraic  numbers  and  relations. 
From  this  time  forth,  symbolic  algebra,  as  we  now  know  it, 
step-by-step,  but  rapidly,  grew  up.  The  advance  in  mathe- 
matics and  mathematical  science  that  soon  followed  is 
almost  incredible.  Thus- the  history  of  mathematics  shows 
two  things,  viz.: 

1.  That  advance  in  mathematical  thought  depends  greatly 
on  the  kind  of  language  employed,  and 

2.  That  the  language  of  modern  symbolic  algebra  is  the  most 
powerful  aid  to  precise  thinking  that  the  world  has  yet  found. 

Every  civiUzed  race  uses  this  language  today.  Of  all 
existing  languages  of  the  world  it  is  best  entitled  to  be  called 
the  universal  language  of  man. 

In  the  second  place,  algebraic  problems  have  definite 
answers,  so  that  the  beginner  may  always  have  a  complete 
check  on  his  thinking  during  the  apprenticeship-period 
while  he  is  necessarily  somewhat  doubtful  about  its  relia- 
bility. On  the  other  hand,  the  problems  of  life  have  no 
answers,  or  the  answers  are  of  the  general  nature  of  success 
or  failure  in  one's  enterprises.  With  the  latter  there  is  no 
chance  to  go  back  and  correct  errors  before  the  errors  have 
resulted  fatally.  This  is  a  strong  reason  why  algebra  is  a 
good  early  training  in  problem-study  and  problem-strategy. 
We  can  do  hard  things  by  virtue  of  the  power  and  skill 
acquired  in  doing  similar,  but  easier,  things. 


'4     "  ELEMENTARY  ALGEBRA 

ALGEBRA  NOT  CREATED  FOR  A  MERE  SCHOOL  DISCIPLINE 

It  thus  appears  that  algebra  was  not  created,  as  pupils 
are  sometimes  prone  to  think,  merely  as  a  severe  discipline 
for  school  boys  and  girls.  Algebra  was  formed  through 
the  united  efforts  of  a  long  succession  of  scientific  men  to 
devise  a  tool  and  technique  for  solving  the  problems  of 
science  that  arose  from  age  to  age  —  problems  that  no 
known  subject  or  device  could  conquer.  It  was  created  as  a 
necessity  to  win  even  the  little  scientific  knowledge  the  race 
acquired  from  age  to  age.  After  algebra  had  revealed  the 
desired  solutions,  sometimes  another  mathematical  subject 
was  found  capable  of  yielding  a  solution  also,  but  algebra 
was  usually  the  pioneer,  and  it  is  only  rarely  that  any  science 
furnishes  easier  and  more  reliable  ways  of  solving  problems 
than  algebra.  To  be  ignorant  of  algebra  is  to  be  deprived 
of  the  most  effective  problem-solving  engine  yet  invented. 
Why  not  seize  the  opportunity  to  acquire  some  mastery 
over  this  powerful  tool?  The  beginnings  of  the  subject  are 
easily  within  the  comprehension  of  the  twelve-year  old  boy 
or  girl. 

ALGEBRA  IS  FUNDAMENTAL  TO  ALL  MATHEMATICAL  SCIENCES 

One  of  the  strongest  reasons  for  studying  algebra  is  that  it  is 
fundamentally  necessary  to  so  many  fields  of  higher  scien- 
tific work.  Aside  from  a  little  elementary  geometry,  almost 
no  mathematics  beyond  the  simplest  arithmetic  is  possible 
without  a  knowledge  of  algebra.  To  attempt  to  get  on  in 
mathematics  without  algebra  is  verily  ''to  try  to  walk  without 
feet."  Perhaps  the  most  widely  useful  mathematical 
subject  within  reach  of  high  school  students  is  trigonometry. 
Trigonometry  is  the  science  of  the  triangle,  and  is  made  up 
very  largely  of  compact  practical  rules,  or  laws,  expressed  in 
the  language  of  algebraic  formulas  and  equations.  The 
transformations  of  these  formulas  that  lead  to  the  most 


REASONS   FOR  STUDYING  ALGEBRA  5 

practical  forms  for  calculating  the  parts  and  properties  of 
triangles,  are  all  algebraic  transforniations.  Much  the 
greater  part  of  trigonometry  is  algebraic.  All  mathematical 
subjects  of  collegiate  rank  require  both  algebra  and  trigo- 
nometry. Algebra  is  indispensable  to  work  in  all  branches 
of  mathematics  beyond  elementary  geometry,  and  nearly 
all  of  higher  arithmetic  is  algebra. 

ALGEBRA  IS  MORE  POWERFUL  THAN  ARITHMETIC 

Many  problems  of  ordinary  life  that  are  commonly  solved 
by  arithmetic,  wpuld  be  much  more  simply  handled  by  alge- 
bra if  the  solver  only  knew  algebra.  Even  before  elementary 
arithmetic  of  the  seventh  and  eighth  grades  is  completed  the 
modern  teacher  finds  many  ways  of  simplifying  the  diffi- 
culties through  equational  modes  of  solution.  Old  fashioned 
analyses  are  today  replaced  by  the  use  of  the  equation  by 
all  well-qualified  teachers.  The  use  of  the  equation  is  alge- 
braic. In  a  very  large  percentage  of  the  problems  of  higher 
arithmetic,  long  and  cumbersome  arithmetical  methods  can 
often  be  replaced  by  simple  mental  algebra.  Why  be  con- 
tent with  a  dull  and  bungling  tool  when  a  sharp  and  handy 
one  is  so  easily  available?  The  modern  farmer  would  scoff 
at  the  idea  of  using  the  sickle  rather  than  the  self-binder  to 
harvest  his  wheat-crop. 

PRACTICAL  MEN  REGRET  NOT  KNOWING  ENOUGH 
HIGH  SCHOOL  ALGEBRA 

It  is  a  common  occurrence  for  business  and  professional 
men  who  have  been  out  of  school  twenty  years  or  more,  to 
express  great  regret  that  they  did  not  give  more  attention  to 
their  high  school  mathematics  when  they  were  in  school. 
They  say  many  Of  their  most  important  problems  are  too 
difficult  for  them,  though  admitting  that  if  their  high  school 
mathematics  had  been  well  done,  they  could  now  solve 
many  of  these  problems.     A  prominent  business  man  said 


6  ELEMENTARY  ALGEBRA 

recently:  ''O,  that  I  knew  enough  algebra  to  enable  me  to 
understand  the  formulas  of  Kent's  Engineers'  Pocket-Book, 
to  be  able  to  make  proper  substitutions  in  these  formulas, 
and  to  know  the  meaning  of  the  results!"  It  is  the  weak- 
ness of  their  problem-solving  ability  that  men  of  practical 
affairs  seem  most  to  regret.  These  men  often  contend  that 
much  of  what  they  had  to  study  in  the  high  school  has 
been  of  little  or  no  use  to  them,  but  that  they  could  not  have 
been  given  too  much  mathematics  for  the  work  they  have 
since  had  to  do.  They  tell  us  the  leaders  today  are  not  the 
great  orators  and  charming  talkers  of  a  generation  ago,  but 
the  mathematized  thinkers.  It  is  the  latter,  they  tell  us,  that 
are  carrying  off  the  prizes  of  this  commercial  and  indus- 
trial age. 

Let  boys  and  girls  who  have  not  yet  lost  the  opportunity 
to  profit  by  school-work  in  mathematics  make  this  study 
as  profitable  as  possible  to  themselves,  by  taking  up  the 
fundamental  subject  of  algebra  with  energy  and  determina- 
tion. Dismiss  the  idea,  if  you  hold  it,  that  you  are  studying 
this  subject  as  a  favor  to  your  teacher  or  parents.  Embrace 
and  cherish  the  true  idea  that  you  are  studying  it  for  your 
own  benefit,  to  raise  your  own  efficiency,  and  that  you  are 
only  cheating  yourself  if  you  do  poor  work.  The  chances 
are  many  to  one  that  the  tasks  of  after-life  will  be  found  to 
make  stronger  demands  on  your  problem-solving  ability  than 
algebra  requires.  Do  not  forget  that  algebra  is  in  a  peculiar 
sense  the  subject  which  can  best  develop  and  perfect  ability 
of  this  type.  Therefore  take  up  the  work  vigorously  the 
first  day,  never  relaxing  your  efforts  to  master  the  subject 
until  the  last  lesson  is  learned. 


ELEMENTARY  ALGEBRA 

First  Half- Year 
CHAPTER  I 

NOTATION   IN   ALGEBRA.    THE  EQUATION 

NOTATION 

1.  The  power  of  algebra  is  due  mainly  to  its  language  and 
its  symbols.  You  have  already  made  some  start  with  this 
language,  for  everything  you  have  correctly  learned  about 
the  language  and  the  symbols  of  arithmetic  holds  good  also 
in  algebra.  But  because  algebra  is  a  sort  of  general  arith- 
metic, it  adds  something  to  the  language  and  symbols  of 
arithmetic  and  employs  them  more  generally  than  arith- 
metic does.  Perhaps  the  most  important  things  for  the 
beginner  to  keep  in  mind  from  the  outset  are  that  what  the 
algebraic  language  talks  about  and  what  the  algebraic 
symbols  stand  for  are  numbers  and  number  relations.  Though 
the  book  or  the  teacher  may  talk  about  algebraic  expressions, 
or  quantities,  or  monomials,  or  polynomials,  it  is  important 
to  remember  that  all  these  terms,  and  many  others,  are 
only  other  names  for  numbers. 

Algebra,  like  arithmetic,  treats  of  numbers.  It  adds, 
subtracts,  multiplies,  and  divides  numbers,  raises  them  to 
powers,  and  extracts  their  roots. 

2.  Notation  is  the  method  of  expressing  numbers  by 
figures  or  letters. 

7 


8  ELEMENTARY   ALGEBRA 

In  arithmetic,  numbers  are  represented  by  ten  Arabic 
characters  called  digits  or  figures.     Thus, 

453  =  400+50+3 

The  Number.  The  whole  of  the  number  is  the  sum  of  the 
parts  represented  by  the  several  digits. 

Representing  Numbers.  In  algebra,  numbers  are  repre- 
sented by  figures,  by  letters,  and  by  a  combination  of  both. 

3.  Products.  When  letters  and  figures  are  written  to- 
gether in  algebra,  their  product  is  indicated. 

Thus,  4a  means  4  times  a,  and  7ab  means  7XaXb 

If  a  number  is  the  product  of  two  or  more  numbers, 
those  numbers  are  factors*  of  the  product. 

The  numbers  represented  by  the  digit  and  the  letters  in 
7ab  are  therefore  factors  of  the  whole  number,  7ab. 

4.  Using  Algebraic  Language.  The  expression,  5x  yards, 
means  5  times  the  number  of  yards  represented  by  x. 

Exercise  1 

1.  What  is  meant  by  the  expression,  9a;  quarts?  12y  cents? 
Sn  miles?   8x  square  feet? 

2.  If  n  represents  a  certain  number,  what  does  4w  repre- 
sent?  9n?   6n? 

3.  If  X  represents  the  price  of  a  yard  of  silk,  what  does 
5x  represent?   8a;?   12a;? 

4.  A  boy  bought  8  oranges  at  h  cents  apiece.  How  many 
cents  did  he  pay  for  them? 

5.  What  is  meant  by  the  expression,  6a;  yards?  4a  dollars? 
Sy  bushels?   7x  square  rods? 

*The  word  factor  means  maker.  The  factors  of  a  number  are  its 
makers  by  multiplication. 


NOTATION  9 

6.  If  a  man  works  for  n  dollars  a  day,  how  much  does  he 
earn  in  8  days?     In  x  days? 

7.  If  a  square  is  x  inches  on  each  side,  what 
does  4x  represent?     What  does  xx  represent? 

8.  How  many  square  inches  are  there  in 
a  rectangle  x  inches  long  and  y  inches 
wide? 


9.  If  you  are  c  years  old  today,  how  old  is  your  father 
who  is  three  times  as  old? 

10.  If  n  represents  a  certain  number,  what  represents  6 
times  the  number?   m  times  the  number? 

5.  Algebraic  Signs.  The  signs  of  addition,  subtraction, 
multiplication,  and  division  mean  the  same  as  in  arithmetic. 

6.  Indicating  Multiplication.  Multiplication  is  often  indi- 
cated in  algebra  by  placing  a  dot  between  the  factors.   Thus, 

5XaXc  =  5rtC  =  5*a*c 

7.  Indicating  Division.  Division  is  often  indicated  by 
writing  the  dividend  over  the  divisor  in  the  form  of  a 
fraction. 

Exercise  2 

1.  Indicate  the  sum  of  8  and  7.  Of  x  and  5.  Of  a  and  h. 
Of  X,  y,  and  z.     Of  2a,  36,  and  12. 

2.  A  man  is  n  years  old  today.  How  old  was  he  7  years 
ago?     Eighteen  years  ago? 

3.  If  a  man  has  p  sheep  in  one  field  and  q  sheep  in 
another,  how  many  has  he  in  both  fields? 

4.  What  is  meant  by  the  expression,  7x  feet?  9y  square 
feet?  n+8days? 


10  ELEMENTARY   ALGEBRA 

6.  When  n  represents  an  odd  number,  what  will  represent 
the  next  larger  odd  number? 

6.  What  represents  the  number  of  square 
inches  in  a  rectangle  x  feet  by  y  inches? 


12x 

7.  If  n  represents  an  even  number,  what  will  represent 
the  next  smaller  even  number? 


8.  How  many  square  inches  are  there  in 
a  rectangle  m  yards  long  and  n  inches  wide?  ^^^ 

9.  If  the  sum  of  two  numbers  is  x  and  the  larger  number 
is  y,  what  is  the  smaller  number? 

10.  If  a  rectangular  piece  of  land  is  x  rods  long  and  y 
rods  wide,  what  does  2x-\-2y  represent? 

11.  Indicate  in  two  ways  the  product  of  5  and  x.     Of 
h  and  y.     Of  5,  a,  and  h.     Of  n,  x,  and  3. 

12.  A  man  bought  x  cows  at  $35  apiece  and  had  $85  left. 
How  much  money  did  he  have  at  first? 

13.  Indicate  the  difference  between  a  and  b  when  a  is 
greater  than  h.    When  b  is  greater  than  a. 

14.  A  boy  had  a  cents.     He  earned  b  cents  and  then  spent 
8  cents  for  candy.     How  many  cents  did  he  have  left? 

15.  A  boy  has  m  quarters  and  n  dimes.     What  expression 
represents  the  number  of  cents  he  has? 

16.  What  represents  the  number  of  square  yards  in  a 
ceiling  x  feet  long  and  y  feet  wide? 

17.  What  will  denote  the  number  of  acres  in  a  rectangular 
'  field  L  rods  long  and  W  rods  wide? 

18.  A  man  sold  a  horse  for  b  dollars  and  gained  c  dollars. 
How  much  did  the  horse  cost  him? 

19.  A  man  bought  x  sheep  at  m  dollars  a  head  and  y  lambs 
at  n  dollars  a  head.     What  did  all  cost  him? 


THE  EQUATION  U 

20.  A  boy  bought  n  apples  at  x  cents  apiece  and  sold 
them  at  y  cents  apiece.     If  he  gained,  what  was  his  gain? 

Since  5  times  any  number +4  times  the  same  number  =  9  times  that 
number,  5a+4o  =  9a,  also 

9x+ix  =  lSx  4a-j-a  =  5a  4n-\-2n-\-n  =  7n 

21.  If  a  man  gets  x  dollars  for  corn  and  4a;  dollars  for 
wheat,  how  much  does  he  get  for  both? 

Since  9  times  any  number  — 3  times  the  same  number  =  6  times  that 
number,  96  —  36  =  66,  also 

I0a-Sa  =  7a  5x-x  =  4x  86-36-6  =  46 

22.  If  the  larger  of  two  numbers  is  8s  and  the  smaller  is  3s, 
what  represents  the  difference   between  the  two  numbers? 

23.  A  has  n  sheep.     B  has  twice  as  many  as  A,  and  C  has 
twice  as  many  as  A  and  B.     How  many  have  all? 

24.  If  a  set  of  harness  costs  x  dollars,  a  carriage  3a;  dollars, 
and  a  horse  5x  dollars,  what  do  all  cost? 

25.  If  a  man  has  5y  dollars  and  spends  x  dollars  for  a 
suit  of  clothes,  how  many  dollars  has  he  left? 

THE    EQUATION 

8.  The  sign  of  equality  is  = .    It  indicates  that  the  numbers 
between  which  it  is  placed  are  equal. 

The  expression   9.t  — 5  =  2x+9,  means  that  the  difference  between 
9x  and  5  is  the  same  as  the  sum  of  2x  and  9. 

9.  An  equation  is  an  expression  of  equality  between  two 
equal  numbers.     Thus, 

8+7  =  5X3  8a:  +  6  =  3x+36  4n+27i  =  54 

10.  The  first  member  of  an  equation  is  the  number  on  the 
left  of  the  sign.  The  second  member  is  the  number  on 
the  right. 


12 


ELEMENTARY   ALGEBRA 


7H8 


-11.  The  equation  expresses  balance  of  values  just  as.  the 
horizontal  position  of  the  bar  of  the 
balances  shows  balance  of  weights. 
To    put    =    between   two    number 
expressions   is  to  say   that   if   the 
numbers  were  weights  and  the  ex- 
pressions in  the  two  members  were  7+8  =  5X3 
represented  by  proper  weights,  one  in  each  pan,  the  balance- 
bar  would  stand  horizontal. 

12.  The  value  of  any  letter  in  a  number  expression  is  the 
number  or  numbers  which  it  represents. 

If  3  unknown  weights  of  x  lb.  each  in  one  pan  are  balanced 
by  6  weights  of.  3  lb.  each  in  the  other  pan,  we  may  say 
3a:  =18.  Leaving  ^  of  the 
weights  on  each  side  on  the 
pans,  and  removing  the  rest, 
the  bar  will  remain  horizontal, 
or  we  may  say,  x  =  6  lb.  That 
is,  the  bar  can  be  horizontal 
when  the  pans  are  loaded  one 
with  3x  lb.  and  the  other  with 
18  lb.  only  if  a;  =  6  lb. 

But  without  troubling  with  the  balance,  by  merely  apply- 
ing the  division  principle  that  equal  numbers  divided  by 
the  same  number  give  equal  numbers,  to  the  equation  3a;  =  18, 
we  find  x  =  6.     In  this  way  algebra  makes  reasoning  take  the* 
place  of  the  weighing  apparatus. 

In  the  equation,  3x  =  18,  since  3x  means  3  times  x,  3x  and  18  are 
equal  only  when  x  represents  6. 

In  the  equation,  3n+2n  =  35,  since  Sn-{-2n  is  5n,  3n+2n  and  35  are 
equal  only  when  n  represents  7;  or  think  thus: 


3a;  =  18 


3n 


2n 


I,  whence  n  =  7. 


35 


THE  EQUATION  13 

Solve  these  exercises  mentally: 

1.  If  7a;  =  28,  what  is  the  value  of  x?  What  is  the  value 
of  n  in  the  equation,  3n+n  =  24? 

2.  If  2a;+3x  =  27,  what  is  the  value  of  x?  What  is  the 
value  of  s  in  the  equation,  3s  — s  =  17? 

13.  The  Unknown  Number.  Any  letter  whose  value  in 
an  equation  is  to  be  found,  is  called  an  unknown  number. 

14.  Solving  an  equation  is  the  process  of  finding  the  value 
of  the  unknown  numbers.  All  changes  in  solutions  of 
equations  are  based  on  simple  statements,  called  axioms. 

AXIOMS 

15.  An  axiom  is  a  statement  the  truth  of  which  is  so 
evident  that  it  may  be  accepted  without  proof. 

Addition  Axiom. — //  the  same  number  or  equal  numbers  are 
added  to  equal  numbers,  the  sums  are  equal. 

If  x  =  a,  x+5=a+5,  and  if  x  =  a  and  7/  =  3,  x-{-y  =  a+3 

Subtraction  Axiom. — //  the  same  number  or  equal  numbers 
are  subtracted  from  equal  numbers,  the  remainders  are  equal. 
If  x  =  y,  x  —  7  =  y  —  7,   and  if  x.  =  y  and  a  =  5,  x—a  =  y  —  5 

Multiplication  Axiom.—//  equ,al  numbers  are  multiplied  by 
the  same  number,  or  by  equal  numbers,  the  products  are  equal. 
If  2n  =  4,  4n  =  8,  and  if  n  =  7  and  x  =  3,  nx  =  3X7 

Division  Axiom. — //  equal  numbers  are  divided  by  the  same 
number  or  by  equal  numbers  {except  zero),  the  quotients  are 
equal. 

If  3a;  =  15,  X  =  5,  andif  a:  =  12and?/  =  3,    -  =^  =  4 

Comparison  Axiom. — Numbers  that  are  equal  to  the  same 
number,  or  to  equal  numbers,  are  equal  to  each  other. 
If  a+6  =  4  and  x—y  =  4,  a-\-b=x—y 


14  ELEMENTARY  ALGEBRA 

In  algebra  when  the  reason  for  a  change  in  an  equation  is 
asked,  the  pupil  is  expected  to  quote  or  to  cite  an  axiom  that 
justifies  the  change. 

16.  Give  the  reason  for  the  conclusion  in  each  of  the 
following : 

1.  a;  =  7and?/  =  4;  thena;+2/  =  ll 

2.  c  =  21  and  d  =  Q;  then  c  —  d  =  15 

3.  a  =  xand6  =  3;  then  a  — 6  =  x  — 3 

4.  c  =  2n;  then  10c  =  20n 

5.  d  =  S2;  then-  =  4 

8 

6.  x  —  7  and  y  =  7;  then  x  =  y 

7.  Sy  =  27;  then  ?/  =  9 

8.  If  n  =  5;  then  8n  =  40 

9.  Ifm  =  9andn  =  4;  then  mn  =  36 

TTl 

10.  If  m  =  28  andn  =  4;  then— =  7 

n 

11.  If  a;-3  =  5;  thena:  =  8 

12.  If  —  =  5;  thena  =  80 

16 

13.  Ifmn  =  7n;  then  m  =  7 

Exercise  3 

1.  Solve  8x-3x+2x-a:  =  30. 

Hx-3x-\-2x-x  =  S0 

6a;  =  30 

By  the  division  axiom,  x  =  5 

Checking,  40-15+10-5  =  30 

or  30  =  30 

Always  check  or  test  the  value  of  the  unknown  number  after  it  is 
found,  by  substituting  it  for  the  unknown  number  in  the  given  equation. 


THE  EQUATION  15 

Solve  and  check: 

2.  Qx-2x+3x  =  49  3.  5n-2n+4n-n  =  48 

4.  5s+6s-3s  =  48  ■                      5.  9a-3a-2a+a  =  45 

6.  Sy-^y-2y  =  24:  7.  86+76-6-46  =  55 

8.  7x-\-2x-3x  =  54:  9.  4n-3n+6n-n  =  72 

17.  Solving  a  problem  is  the  process  of  finding  the  values  of 
the  unknown  numbers  involved  in  the  problem. 

In  arithmetic  the  unknown  numbers  are  found  by  one  or 
more  of  the  fundamental  processes. 

In  algebra  the  unknown  numbers  are  represented  by  letters 
and  their  values  are  found  by  the  use  of  equations. 

Solving  a  problem  in  algebra  involves  three  steps :  notation , 
statement,  solving  an  equation. 


Exercise  4  —  Solving  Problems 

1.  The  sum  of  two  numbers  is  252,  and  the  larger  number 
is  6  times  the  smaller.     Find  the  numbers. 

_-.       .  j     Let  s  =  the  smaller  number; 

'       \ then  6s  =  the  larger  number. 

Hence  s+6s  and  252  are  two  number  expressions,  each  of  which 
represents  the  sum  of  the  two  numbers. 

Statement,  s+6s  =  252 

The  notation  is  the  representation  in  algebraic  symbols 
of  the  unknown  numbers  in  the  problem. 

The  statement  is  the  expression  of  the  conditions  of  the 
problem  in  one  or  more  equations. 

f7s  =  252 
Solving  the  equation,        I   s  =  36 

6s  =  216 


16  ELEMENTARY  ALGEBRA 

To  cheeky  substitute  in  the  statement.     Thus, 
36+216  =  252,  or  252  =  252 

Even  the  statement  itself  may  be  wrong.  To  test  whether  this  is 
the  case,  substitute  in  the  conditions  of  the  problem  itself. 

2.  The  sum  of  two  numbers  is  846,  and  the  larger  number  is 
8  times  the  smaller.     Find  the  numbers. 

3.  Seven  times  a  certain  number  plus  6  times  the  number 
minus  8  times  the  number  equals  175.     Find  the  number. 

18.  Obtaining  Statements  of  Problems.  To  obtain  the 
statement  in  a  problem  is  to  translate  the  conditions  of  the 
problem  into  an  equation. 

DIRECTIONS  FOR   MAKING   STATEMENTS   AND   SOLVING   PROBLEMS 

1.  Let  any  appropriate  letter  represent  one  of  the  unknown 
numbers  to  be  found. 

2.  From  the  conditions  of  the  problem  express,  in  terms  of 
the  same  letter,  the  other  unknown  numbers. 

3.  Find  two  number  expressions  that  represent  the  same  num- 
ber and  place  them  equal,  forming  an  equation. 

4.  Solve  the  equation  and  determine  whether  the  result  satis- 
fies the  conditions  of  the  problem. 

Exercise  5 

1.  One  number  is  5  times  another,   and  the  difference 

between  them  is  48.     Find  the  numbers. 

^       .  jLet      s  =  the  smaller  number; 

'       \then  5s  =  the  larger  number. 
Hence,  5s  — s  and  48  are  two  number  expressions,  each  of 

which  represents  the  difference  between  the  numbers. 
Statement,  5s  —  s  =  48 

Solving  this  equation,  s  =  12  and  5s  =  60 

Checking,  60-12=48,  or  48  =  48 


THE   EQUATION  17 

2.  A  is  six  times  as  old  as  B,  and  the  difference  between 
their  ages  is  75  years.     Find  B's  age. 

3.  In  a  school  of  855  pupils  there  are  twice  as  many  girls 
as  boys.     How  many  girls  are  there? 

4.  A  earned  five  times  as  much  as  B.  If  B  earned  $648 
less  than  A,  how  much  did  both  together  earn? 

5.  The  length"  of  a  rectangle  is  3  times  its  width,  and  the 
perimeter  is  224  inches.     Find  the  dimensions. 

19.  Letters  Represent  Numbers.  In  solving  problems, 
always  let  the  letter  represent  some  number.  It  must  not 
represent  money,  but  a  number  of  dollars  or  cents;  not  time, 
but  a  number  of  days  or  hours;  not  weight  but  a  number  of 
pounds  or  ounces;  not  distance,  but  a  number  of  miles,  rods, 
or  other  units  of  measure. 

Exercise  6  — Problems 

1.  A  horse,  carriage,  and  harness  cost  $450.  The  carriage 
cost  3  times  as  much  as  the  harness,  the  horse  twice  as  much 
as  the  carriage.     Find  the  cost  of  each. 

I     n     I  3«  I  6re  i 

I  I ! I I I  I I I ! 


450 

Let  n  =  the  number  of  dollars  the  harness  cost; 
then  3n  =  the  number  of  dollars  the  carriage  cost; 
and       6ri  =  the  number  of  dollars  the  horse  cost. 

Hence  n+3n+6n  and  450  are  two  number  expressions, 
each  of  which  represents  the  cost  of  all. 

n+3n+6n  =  450 

2.  One  number  is  9  times  another,   and  the  difference 
between  them  is  624.     Find  the  numbers. 


624 


18 


ELEMENTARY  ALGEBRA 


3.  A  has  twice  as  many  sheep  as  C,  and  B  has  4  times  as 
many  as  C.     If  all  have  665,  how  many  has  B? 


C's 


A' 8 


665 

4.  A  house  and  lot  cost  $7250,  the  house  costing  4  times 
as  much  as  the  lot.     Find  the  cost  of  each. 

I         lot         I  co,it  of  house  , 


7250 


5.  If  twice  a  number  is  added  to  six  times  the  same  number, 
the  sum  is  192.     Find  the  number. 


192 


6n 


6.  The  sum  of  the  ages  of  father  and  son  is  96  years,  and 
the  difference  between  their  ages  is  twice  the  son's  age. 
What  is  the  father's  age? 


father's  age 


2  times  sons  age 


7.  A  rectangle  formed  by  placing  two 
equal  squares  side  by  side  has  a  perim- 
eter of  270  feet.  Find  the  side  of  each 
square  and  the  area  of  th^  rectangle. 

8.  If  two  rectangles  of  the  same  width  and  twice  as  long 
as  wide  are  placed  end   to  end,  the 
perimeter  of  the  rectangle  formed  is 
180  inches.     Find  their  dimensions. 


2w 


2w 


9.  One   number   is   4   times   another,  and  4  times  their 
difference  is  576.     Find  the  numbers. 

10.  A  man  sold  a  horse  and  carriage  for  $340,  receiving  3 
times  as  much  for  the  horse  as  for  the  carriage.  How  much 
did  he  get  for  the  carriage? 


THE   EQUATION  19 

11.  The  sum  of  two  numbers  is  322,  and  their  difference  is 
5  times  the  smaller.     Find  the  larger  number. 

12.  A,  B,  and  C  own  840  sheep.  A  owns  3  times  as 
many  as  B,  and  C  owns  twice  as  many  as  A  and  B  together. 
How  many  do  A  and  C  together  own? 

13.  A's  age  exceeds  B's  by  3  times  B's  age,  and  the  sum  of 
their  ages  is  75  years.     Find  A's  age. 

14.  In  a  mixture  of  228  bushels  of  corn  and  oats  there  are 
twice  as  many  bushels  of  corn  as  of  oats.  How  many  bushels 
of  oats  are  there  in  the  mixture? 

15.  A  number  increased  by  3  times  itself,  4  times  itself,  and 
5  times  itself  is  650.     Find  the  number. 

16.  A  man  sold  some  lambs  at  $3  a  head  and  three  times 
as  many  sheep  at  $5  a  head,  receiving  $324  for  all  of  them. 
How  many  of  each  did  he  sell? 

17.  The  length  of  a  rectangle  is  4  times  its  width,  and  the 
perimeter  is  280  yards.     Find  the  dimensions. 

18.  A,  B,  and  C  own  600  acres  of  land.  B  owns  3 
times  as  many  acres  as  A,  and  C  owns  half  as  many  acres 
as  A  and  B  together.     How  many  acres  have  B  and  C? 

19.  A  merchant  paid  $50  for  two  pieces  of  silk  of  equal 
value,  paying  80ff  a  yard  for  one  piece  and  $1.20  a  yard 
for  the  other.     How  many  yards  were  in  each  piece? 

20.  Two  equal  rectangles  whose  length  is  3  times  the  width, 
if  placed  end  to  end,  form  a  rectangle  whose  perimeter  is 
196  inches.     Find  the  length  of  each  rectangle. 


CHAPTER   II 

POSITIVE   AND   NEGATIVE   NUMBERS. 
DEFINITIONS 

POSITIVE  AND  NEGATIVE  NUMBERS 

20.  Numbers  of  Arithmetic.  The  only  relation  of  numbers 
considered  in  arithmetic  is  the  relation  of  size. 

A  boy  starts  from  0,  takes  12  steps  toward  the  right,  then 
turns,  and  takes  7  steps  toward  the  left.  How  far  is  he  then 
from  the  starting-place,  0? 

In  arithmetic  we  would  solve  this  problem  thus : 
12-7  =  5 

But  suppose  after  taking  12  steps  to  the  right  and  turning 
back,  he  had  taken  20  steps  toward  the  left.  Where  would 
he  then  be  with  regard  to  the  starting-point? 

I   ...    I   I    I    ...    I    I    I    I    I   I    I    I   I    I    I   I    I    I    I    I    ..   I    I    I 

-15  -10  -5-4-3-2-1    OM+2+3+4+5  H-IO  +15 

An  Algebraic  Scale 

We  know  that  in  arithmetic  we  cannot  subtract  20 
from  12.  Still  by  using  the  algebraic  scale  above,  we  can 
easily  solve  the  problem,  and  learn  that  the  boy  will  be  8 
steps  to  the  left  of  the  starting-point,  0.  If  we  agree  that 
the  sign  — ",  instead  of  meaning  ''subtract"  shall  mean  "go 
leftward,''  we  may  write: 

12-20= -8. 

It  will  be  more  complete  also  to  agree  that  the  sign,  -f, 
instead  of  always  meaning  "add,"  as  it  did  in  arithmetic, 
may  mean  also  "go  rightward'' ;  hence  we  write: 

+  12-20= -8, 
which  means  *'12  steps  right  ward,  followed  by  20  steps  left- 
ward, leaves  one  8  steps  left  of  the  starting-point." 

20 


POSITIVE  AND  NEGATIVE  NUMBERS  21 

This  is  what  we  do  in  algebra,  thus  making  it  possible 
to  solve  numerous  problems  that  cannot  be  solved  by  arith- 
metic. Hence,  to  learn  algebra  is  to  add  greatly  to  our 
problem-solving  power. 

21.  Such  numbers  as  +12,  —20,  and  —8  are  called 
directed  numbers,  or  signed  numbers,  and  the  +  or  the  — 
is  just  as  much  a  part  of  the  number  as  is  the  12,  the  20,  or 
the  8. 

There  is  now  nothing  impossible  about  such  a  problem  as: 

+6-15  =  ? 

By  referring  to  the  algebraic  scale,  tell  what  the  answer  is. 

Thus,  in  algebra  the  signs  +  and  —  may  mean  add  and 
subtract,  or  the  direction,  or  kind,  or  quality  of  the  number, 
i.e.,  they  are  verbs  or  adjectives. 

22.  Algebraic   Nimibers  have   Opposite    Qualities.     We 

shall  learn,  that  many  numbers  in  their  relation  to  each 
other  are  opposite  in  quality. 

Gains  and  losses,  owns  and  owes,  dates  before  and  after, 
and   distances  in   opposite   directions   serve   as    examples. 

In  algebra,  numbers  are  considered  with  reference  to  the 
two  relations  of  size  and  opposite  quality. 

ILLUSTRATIONS 

If  a  man  makes  $4500  one  year  and  loses  $2500  the  next  year,  his 
net  gain  for  the  two  years  is  $2000. 

If  a  merchant's  assets  are  $14,000  and  his  liabilities  are  $6000,  he 
is  really  worth  only  $8000. 

Numbers  are  of  opposite  quality;  therefore  in  combining 
them,  the  smaller  number  united  with  an  equal  part  of  the 
larger  number  gives  zero. 

23.  Positive  and  Negative  Numbers.  To  describe  the 
opposite  quality  of  numbers,  the  terms  positive  and  negative 


22  ELEMENTARY  ALGEBRA 

are  used  in  algebra,  and  the  quality  of  a  number  is  denoted 
by  the  sign  +  or  — . 

The  sign  +  before  a  number  denotes  that  it  is  positive,  and 
the  sign  —  that  it  is  negative,  as  +5,  —6. 

24.  The  absolute  value  of  a  number  is  the  number  of  units 
in  it,  independent  of  their  quality. 

The  absolute  value  of  +9  is  9. 
The  absolute  value  of  —8  is  8. 

Exercise  7 

Let  us  consider  distance  north  from  a  certain  point  as  posi- 
tive and  distance  south  as  negative. 

1.  If  a  man  walks  north  12  miles  one  day  and  north  13 
miles  the  next  day,  what  is  the  result? 

2.  If  a  man  walks  south  11  miles  one  day  and  south  10 
miles  the  next  day,  what  is  the  result? 

3.  If  a  man  walks  north  14  miles  one  day  and  south  10 
miles  the  next  day,  what  is  the  result? 

4.  If  a  man  walks  north  10  miles  one  day  and  south  15 
miles  the  next  day,  what  is  the  result? 

6.  If  a  man  walks  south  14  miles  one  day  and  north  11 
miles  the  next  day,  what  is  the  result? 

6.  If  a  man  walks  south  10  miles  one  day  and  north  17 
miles  the  next  day,  what  is  the  result? 

You  have  probably  answered  these  six  questions  as  follows:  He  is 
25  miles  north  of  the  starting-point;  21  miles  south;  4  miles  north; 
5  miles  south;  3  miles  south;  7  miles  north. 

Here  are  the  algebraic  solutions  of  the  six  problems.  Tell  how  each 
result  is  obtained  and  what  it  represents. 

+  12  -11  +14  +10  -14  -10 

+  13  -10  -10  -15  +11  +17 

+25  -21  +4  -5  -3  +7 


POSITIVE  AND  NEGATIVE  NUMBERS  23 

The  results  of  uniting  these  positive  and  negative  numbers 
show  the  following  principles : 

25.  The  sum  of  two  numbers  with  like  signs  is  the  sum  of 
their  absolute  values  with  the  common  sign  prefixed. 

26.  The  sum  of  two  numbers  with  unlike  signs  is  the  differ- 
ence between  their  absolute  values  with  the  sign  of  the  number 
having  the  greater  absolute  value  prefixed. 

Exercise  8 

Applying  these  principles,  write  the  sums  in  the  following 
examples,  giving  each  its  proper  sign: 

+23  +33  -31  -41  +29  +19 

+  15  -14  -16  +17  -14  -37 


+75 

+83 

-67 

+43 

-28 

-73 

+68 

-38 

-49 

-82 

+74 

+37 

+85 

+39      * 

-49 

+93 

-65 

-34 

+78 

-75 

-68 

-45 

+29 

+73 

27.  Double  Meaning  of  +  and  — .  Thus  it  appears  that 
the  signs  +  and  —  are  used  in  algebra  to  denote  quality  of 
numbers  as  well  as  to  denote  operations. 

Exercise  9 — Problems  with  Positive  and  Negative  Numbers 

Assign    quahty    to    the    numbers    in    these    problems, 
solve  them  algebraically,  and  interpret  the  results : 

1.  A  man's  property  amounts  to  $18,750  and  his  debts  to 
$23,250.     Find  his  net  debt  or  property. 

2.  A  merchant  gains  $2365  one  year  and  loses  $1790  the 
next  year.     Find  the  net  gain  or  loss. 

3.  If  a  man  travels  east  58  miles  one  day  and  west  73  miles 
the  next  day,  what  is  the  net  result? 


24  ELEMENTARY   ALGEBRA 

4.  A  man's  annual  income  is  $3675  and  his  expenses  $2395. 
How  much  does  he  save  annually? 

5.  If  a  ship  sails  north  53  miles  one  day  and  south  39 
miles  the  next  day,  what  is  the  net  result? 

6.  A  real  estate  dealer  gains  $1465  on  one  sale  and  $2375 
on  another.     Find  the  result  of  both  sales. 

7.  Draw  a  line  representing  a  thermometer  scale;  mark  the 
zero  point,  24°,  and  — 12°.  What  is  the  difference  between 
the +24°  and  the  -12°? 

8.  If  the  weight  of  a  stone  is  regarded  as  positive,  what 
would  represent  the  weight  of  a  balloon? 

9.  If  a  speculator  makes  $2765  one  month  and  loses  $2875 
the  next  month,  what  is  his  net  gain  or  loss? 

10.  If  a  stone  weighs  34  pounds,  and  a  balloon  pulls  upward 
with  a  iorce  of  8  pounds,  what  is  the  combined  weight  of 
both,  if  they  are  fastened  together? 

DEFINITIONS 

28.  A  system  of  notation  is  a  system  of  symbols  by  means 
of  which  numbers,  the  relations  between  them,  and  the 
operations  to  be  performed  upon  them  can  be  more  concisely 
expressed  than  by  the  use  of  words. 

29.  Algebraic  notation  is  the  method  of  expressing 
numbers  by  figures  and  letters. 

30.  An  algebraic  expression  is  the  representation  of  any 
number  in  algebraic  notation. 

31.  A  term  is  a  number  expression  whose  parts  are  not 

separated  by  the  sign  +  or  — ,  thus, 

bx 
2aX46,     3a6,     xy^     box,     and    — 


DEFINITIONS  '       25 

32.  A  monomial  is  an  expression  oi  ^one  term.  A  poly- 
nomial is  an  expression  of  two  or  7nore  terms,  as, 

2a+46-3c-5d 
The  signs  +  and  —  between  the  terms  of  a  polynomial 
may  be  regarded  as  signs  of  operation  or  of  quality. 

When  monomials  and  the  first  term  of  a  polynomial^  are  ^itten 
without  any  sign  before  them,  they  are  positive. 

33.  A  binomial  is  a  polynomial  of  two  terms.  A  trinomial 
is  a  polynomial  of  three  terms. 

34.  A  coefficient  of  a  term  is  any  factor  of  the  term  which 
shows  how  many  times  the  other  factor  is  taken  as  an 
addend.     Thus,  ' 

4n  =  n-fnH-n4-n  4ax  =  ax-^ax-\-ax-\-ax 

Coefficients  are  distinguished  as  numerical  or  literal,  according  as 
they  are  expressed  in  figures  or  letters. 

In  the  two  terms  above,  4  is  the  numerical  coefficient. 

Any  other  factor  of  4ax  may  be  regarded  as  the  coefficient  of  the 
product  of  the  remaining  factors. 

Observe  that  Aa-\-a  =  a-\-a-\-a-\-a-\-a  =  ba 

This  shows  that  when  no  numerical  coefficient  is  expressed, 
the  numerical  coefficient  is  considered  to  be  1. 

35.  Similar  terms  are  terms  which  do  not  differ,  or  which 
differ  only  in  their  numerical  factors,  as, 

bxy,  xy,  and  Sxy;     Sab  and  5a6;     or  4ax,  ax,  and  7 ax 

36.  Dissimilar  terms  are  terms  that  are  not  similar,  as 

4ab,  ax,  36c;     Sac,  ixy;     2xy,  xz,  3yz 

37.  Partly  Similar  Terms.  Terms  that  have  a  common 
factor  are  said  to  be  partly  similar,  or  similar  with  respect 
to  that  factor. 

Thus,  ax,  ix,  and  bx  are  similar  with  respect  to  x;  and  5xy,  axy, 
bxy,  and  4cxy  are  similar  with  respect  to  xy. 


26 


ELEMENTARY  ALGEBRA 


38.  The  value  of  an  algebraic  expression  is  the  number  it 
represents  when  some  particular  value  is  assigned  to  each 
letter  in  the  expression. 

Substitute  1  for  a,  2  for  6,  3  for  c,  4  for  d,  in  the  following 
expression  and  simplify  the  result : 

2ah  +   36c   +  5cd   -  4hd  = 
2-l-2+3-2-3+5-3-4-4-2-4  = 
4     +18+60-32    =50 


Exercise  10 


Find  the  value  of  each  of  the  following  expressions  when 
,  a  =  l,  6  =  2,  c  =  3,  c?  =  4,  e  =  0,  m  =  |,  n  =  J: 


1.  hcd—dn—^a-^-Qam 

3.  Qh  —  adm-{-5hc  —  Qn 

5.  hcd  —  5e  —  4:m-jrScn 

7.  Sa-\-Qmn-2b-\-bcd 

9.  5cd  —  Sm-{-9a  —  Qcn 

11.  4bc-\-7d-9n+7ab 

13.  cdm-{-Scn-\-ab  —  7e 

15.  9n-\-cdm-\-de  —  2ab 


2.  6am+9a+26c-3n 

4.  4ad-6n+2d(m-26 

6.  Sb-4am+9bn-2a 

8.  Sad—7e  —  bm-\-Qdn 

10.  7a-\-9bd-Sm-\-9an 

12.  56d+ac— 46m+6n 

14.  66n+5e  — 6m+8ad 

16.  cd  —  acn-\-Sm-\-Sem 


CHAPTER  III 

ADDITION 
ADDITION   OF   MONOMIALS 

39.  Addition  is  the  process  of  uniting  two  or  more  numbers 
into  one  number. 

40.  The  addends  are  the  numbers  to  be  added;  the  sum  is 
the  number  obtained  by  addition. 

41.  To  Add  Similar  Terms.     In  adding  5-6  and  3 '6  in 

arithmetic,  the  two  products,  which  are  30  and  18,  are  found 
and  then  added. 

Since  5  times  6  plus  3  times  6  is  8  times  6,  they  may  be 
added  also  by  adding  the  coefficients  of  6,  thus 
5.6+3-6  =  8-6 

42.  Adding  Indicated  Products.  Algebraic  terms,  which 
are  indicated  products,  can  be  united  into  one  term  only  by 
the  latter  method.     For  example: 

1.  A  school  hall  is  I  yards  long.  I  go  through  it  6  times  on 
Monday  and  14  times  on  Tuesday.  How  many  yards  do  I 
travel  through  the  hall  on  both  days? 

Monday,     6/  yards 

Tuesday,  14'^  yards 

Both  days,  201  yards 

2.  The  tickets  for  an  entertainment  were  t  cents  each. 
George  sold  34  and  Mary  28  tickets.  Find  the  total  receipts 
from  the  sales  of  George  and  Mary. 

George,  34^  cents 
Mary,  28^  cents 
Both,       Q2t  cents 

27 


28  ELEMENTARY  ALGEBRA 

ADDING   SIMILAR   TERMS 

43.  The  sum  of  two  similar  terms  is  the  sum  of  their  coeffi- 
cients with  the  common  letters  affixed. 

Whether  the  terms  have  Uke  or  unhke  signs,  the  sum  of 
the  coefficients  is  found  by  §§  25  and  26. 

Exercise  11 

Give  at  sight  the  sum  of  each  of  the  following : 

1.  4-3         2.       4a  3.       8a;  4.   -76  5.   -3c 

5-3  -5a  -3x  b  -5c 


6.  8-5 

7. 

9a 

8. 

-9x 

9. 

96 

10. 

-6c 

6-5 

-3a 

4a; 

-6 

-4c 

11.  6-7 

12. 

2x 

13. 

-46 

14. 

n 

16. 

-3c 

7-7 

-7x 

56 

-4n 

-7c 

16.  5a         17.   -7x  18.       26  19.   -  n         20.   -8c 

6a  6a;  —66  6n  —6c 


21.     a         22.       9a;  23.   -76  24.   -8n  25.   -4c 

7a  —3a;  86  n  —9c 


26.  5a         27.   -9a;  28.       96  29.   -   n         30.   -7c 

9a  8a;  -26  4n  -5c 


31.  7a         32.       47/  33.   -66  34.   -7n  35.   -6c 

8a  -9y  76  n  -9c 

44.  Rule. —  Find  the  algebraic  sum  of  the  coeffix^ients,  and 
to  that  result  affix  the  comm^on  letters. 


ADDITION  OF  MONOMIALS  29 

45.  Fundamental  Laws.  There  are  two  fundamental  laws 
of  addition  which  it  will  be  well  to  notice  here.  They  are 
known  as  the  law  of  order,  or  the  commutative  law;  and  the 
law  of  grouping,  or  the  associative  law. 

46.  Law  of  Order.  The  sum  of  two  or  more  numbers  is  the 
sam£  in  whatever  order  they  are  added. 

It  is  evident  that: 

8+6+4=6+4+8=4+8+6 
for  each  member  of  this  equality  is  the  same  number. 
This  law  is  represented  as  follows : 

a+b+c  =  b+c+a  =  c+a+b  =  a+c+b,  etc. 

47.  Law  of  Grouping.  The  sum  of  several  numbers  is  the 
same  in  whatever  manner  they  are  grouped. 

Thus,  8+6+4  denotes  that  6  is  to  be  added  to  8,  and  4 
added  to  the  result;  that  is,  8+6+4=  (8+6)+4. 
By  the  law  of  order, 

8+6+4=6+4+8=4+8+6 
Therefore, 

8+6+4=  (8+6)+4=  (6+4) +8=  (4+8) +6 
This  law  is  represented  as  follows : 

a+b+c=  (a+b)+c=  (b+c)+a=  (a+c)+b 

Adding  Several  Positive  and  Negative  Terms.  The  addition  of 
several  similar  terms  with  unlike  signs  is  based  on  the  associative  law. 
By  this  law,  the  positive  terms  are  grouped  together  and  added,  the 
negative  terms  are  grouped  together  and  added,  and  the  two  sums  then 
united. 

Exercise  12  —  Adding  Similar  Terms 

Give  the  sum  of  each  of  the  following  : 

1.  2a           2.   ~4:X            3.       56  4.   -3/i  5.   -   c 

a                  2x                -lb  6/1  -7c 

3a                —  X                    36  — 4n  •    —9c 


30 


ELEMENTARY   ALGEBRA 


6.  a 

7. 

Qx 

8. 

-96 

9. 

2n 

10.  -8c 

8a 

-9x 

36 

-4n 

—  c 

2a 

X 

76 

Sn 

-9c 

11.  a 

12. 

-7x 

13. 

56 

14. 

Qn 

15.  -3c 

5a 

4x 

-66 

-9n 

9c 

a 

-Qx 

96 

2n 

-6c 

16.  5a 

17. 

Sx 

18. 

-  6 

19. 

Qn 

20.  -9c 

a 

-Sx 

-  6 

—  n 

4c 

7a 

2x 

86 

Sn 

-8c 

ADDING  DISSIMILAR  TERMS 

48.  Dissimilar  terms  cannot  be  united  into  one  term. 
The  addition  can  only  be  indicated  by  writing  them  in  suc- 
cession in  any  order,  each  preceded  by  its  own  sign,  as  here 
shown : 

3ac  5a  —36c 

—   6c  —26  — 4ac 

2bd  -3a  -26 


dac-bc-{-2bd 


2a -26 


-4ac-36c-26 


We  write  a  positive  term  first,  if  there  is  one.     If  all  the  terms  are 
negative,  any  one  of  them  may  be  written  first. 


Exercise  13 
Give  at  sight  the  sum  of  each  of  the  following : 


1.  3a 

2.   6 

,    3.  -2x 

4.  -2n 

6.  -5c 

2x 

-2c 

y 

-Sx 

—  c 

6.  4a 

7.   2a 

8.-71 

9.   2a 

10.  5x 

Sx 

-  6 

-3a: 

-  6 

-Qy 

a 

—  c 

4n 

-5c 

-4x 

ADDITION  OF  MONOMIALS  31 

11.    2a           12.      ^x           13.-   X           14:. -In  15.    •  2a 

*    6                 —4a;                 —32/                 —  n  —    c 

5a;                —  5x                   4x                   4n  —5c 


16. 

4a 

17.   4a; 

18.  -36 

19. 

3a 

20.  -7c 

26 

-2y 

66 

—  c 

-9c 

a 

—    X 

-46 

-7n 

-8c 

21. 

la 

22.  -5a; 

23.   46 

24. 

-Sn 

25.  -6c 

a 

32/ 

n 

7n 

4c 

4a 

a; 

-Qx 

-5n 

-5c 

26. 

3a; 

27.  -2a; 

28.   76 

29. 

-2n 

30.  -5a 

a 

5a; 

-4c 

Sn 

-26 

5a 

-3a; 

6 

-6n 

-42/ 

31. 

a 

32.  -5a; 

33.  -3a 

34. 

3n 

35.  -  a 

3n 

3a; 

6 

-9/1 

-5c 

a; 

-7a; 

—  a 

7n 

—  c 

36. 

4a 

37.   X 

38.  -66 

39. 

-4n 

40.    c 

a 

-  y 

-  6 

2a 

9c 

2a 

4z 

-76 

n 

—  c 

41. 

y 

42.  -5a; 

43.   6a 

44. 

n 

45.  -  a 

7x 

3a; 

-  6 

-3n 

-3c 

x 

-7x 

2n 

n 

-4a 

46. 

4a 

47.   36 

48.  -7y 

49. 

5c 

50. -7a; 

a 

-56 

y 

6 

3a; 

6a 

-  6 

Qy 

-2c 

—  X 

a 

96 

-42/ 

26 

6a; 

32  ELEMENTARY   ALGEBRA 

Simplify  the  following: 

51.  4a+2a+a+5a  52.  3n+8n+n+2n+6n 

53.  2x+x-lx+^x  54.  56-26-66+6+96 

55.  5c-6c+c+4c  56.  Qiy-y-\-^y-1y-{-by 

57.  7a— 3a  — a  — 2a;  58.  7n+5yi  — n— 4n+3a 

59.  8a;-4a;-3a;-2/  60.  46+96+76-86-6 

61.  6i/+8x-9i/-5a:  62.  6a-76-4a+36+a 

ADDITION   OF  POLYNOMIALS 

49.  Addition  of  polynomials  proceeds  much  as  addition  of 
monomials,  as  the  two  following  illustrations  show: 

1.  The  stairway  of  a  school  has  3  flights,  of  a,  6,  and 
c  steps,  respectively.  A  boy  goes  up  and  down  the  stairway 
3  times  on  Monday,  5  times  on  Tuesday,  4  times  on  Wed- 
nesday, 6  times  on  Thursday,  and  4  times  on  Friday. 
How  many  steps  does  he  take  on  the  stairs  during  the  week? 

Monday,  6a  +  66+  6c  steps 

Tuesday,  10a +  106  + 10c  steps 

Wednesday,  8a  +  86+  8c  steps 

Thursday,  12a +  126  + 12c  steps 

Friday,  8a  +  86+  8c  steps 

Sum,  44a +446+ 44c  steps 

2.  At  a  money-changer's  are  offered  for  exchange: 

At  one  time,     52  marks,  35  francs,  12  pounds; 

At  another,       18  marks,  26  francs,  24  pounds; 

At  another,      22  marks,  15  francs,  18  pounds. 
The  exchange  value  of  a  mark  being  m  cents,  of  a  franc 
/  cents,  and  of  a  pound  /  cents,  find  the  total  exchange  value 
of  the  foreign  currency  in  cents. 

First  time,  52m +35/+ 12i  cents 

Second  time,  18w+26/+24i  cents 

Third  time,  22m  +  15/+18Z  cents 

Sum,  92m+76/+54/  cents 


ADDITION  OF  POLYNOMIALS  33 

50.  To  add  poljmomials,  ivrite  similar  terms  in  a  column 
and  add  each  column,  beginning  at  the  left. 

Thus,  5ah+Sac-2bc+Sbd-{-  5xy  -  7xz 

2ab—  ac  —5bd-{-2xy 

5ac—  be  -\-7xz 

ab  -\-3bc  -4:xy  +6 

Sab+7ac  -2bd+3xij  +6 

51.  A  check  on  algebraic  work  is  another  operation  which 
tends  to  prove  the  first  result  correct. 

52.  Checking  Addition  by  Substitution.  Addition  may  be 
checked  by  substituting  any  number  in  place  of  the  letters 
and  determining  whether  the  sum  of  the  valv£s  of  the  addends 
equals  the  value  of  the  sum. 

The  following  shows  how  addition  of  polynomials  may  be 
checked  by  substituting  1  for  each  letter. 

Work  Check 


5a-96+7c  =  5-9+7  =  3 

a+Sb-Qc  =  1+8-6  =  3 

3a-46+3c  =  3-4+3  =  2 

9a-56+4c  =  9-5+4  =  8 

The  sum  of  the  values  of  the  addends  is  8,  and  the  value  of 
the  sum  of  the  polynomials  is  also  8. 

Observe  that  when  1  is  substituted  for  each  letter,  the 
value  of  each  term  is  the  numerical  coefficient. 

In  checking  or  verifying  algebraic  processes,  any  number 
may  be  substituted  for  each  letter.  To  avoid  large  num- 
bers, it  is  well  to  substitute  small  numbers;  but  substitu- 
ting 1  checks  only  the  coefficients  and  should  not,  in  general, 
be  done. 


34  ELEMENTARY  ALGEBRA 

Exercise  14 

1.  Add4a-3n+2:r,  5n-4a;+5,  -7a-4n+7x,  2a+6n4-6, 
n  — X— 14,  and  5a  — 4n  — 3x+4. 

2.  Add  5b-i-Sc-Qd,  c-2b+Sd,  -M-dc+Sb,  6d-4c-76, 
and  2c-\-d-\-4:b,  and  check. 

3.  Add4a-Sb+5c,2c-2b-\-d,  -4d-8a+76,  3c+4a+3(i, 
and  26  — d— 7c,  and  check. 

4.  Add7x-5i/+32,  31/-8-52;,  -4:y-\-Qz-5x,  Qy-2x-Sz, 
and  42 +32/ +8,  and  check. 

5.  Add    Sax-{-4iby  —  2xy,    5by  —  7xz-\-Qxy,    2ax  —  Sxy  —  dby, 
and  7xz-\-xy  —  by,  and  check. 

6.  Add    2x-4y-{-Sz,    -bz+y-Qx,    3y+z-^5x,    -Sy+Ax 
—42,  and  7y+Sz—4iX. 

7.  Add    7ac  —  an-\-Snx,    5ax-{-4:an  —  Qnx,    2nx  —  San  —  5ac, 
and  an—5ax  —  nx,  and  check. 

8.  Add  5a+66-7c,  4c-36+5,  -2c+5b-Sa,  4c-76-9, 
and  -6+2c+6aH-5. 

9.  Add    5ax-\-3bx  —  2cx,    Zdx—4:ax-\-5cx,    4:cx  —  7bx  —  3dx, 
and  66a;  — Sex + ax. 

10.  Add    8a6  — 66c+4ac,      5ad  —  5ab  —  7ac,      46c  — ad+5ac, 
and  36c  — 3ad— 3a6. 

11.  Add   4an  — 76n+5a6,   46n  — 7ac  — 6an,    3anH-6ac  — 9a6, 
and  4bn  —  ac-\-4ab,  and  check. 

12.  Add    6x-7y+5z,    4y-u-Sz,    -2u-\-Qy-5x,    Az-by 
+4:U  —  x,  and  —Qz-\-2x  —  Su. 

13.  Add  4a6  — 2ac+46c,    —  5ac  — 2a6+66c,  and  ab  —  2ac. 

14.  Add    4a -76 -5c,    Sc-7a-d,     -56+3d-2c,    86-2cZ 
-5+c,  and  46+2c+4a+5. 

15.  Add     5xy-ixz+dyz,     2xz  —  2xy  —  7yz,     3xz—xy-\-9yz, 
and  —Syz+Qxy  —  xz,  and  check. 


CHAPTER  IV 
SUBTRACTION.     SYMBOLS    OF   AGGREGATION 

53.  Subtraction  is  the  process  of  finding  one  of  two  num- 
bers when  their  sum  and  the  other  number  are  known. 

54.  The  minuend  is  the  number  that  represents  the  sum; 
the  subtrahend  is  one  of  the  addends  of  the  minuend. 

55.  The  difference,  or  remainder,  is  the  number  which 
added  to  the  subtrahend  gives  the  minuend. 

SUBTRACTION  OF  MONOMIALS 

1.  A  thermometer  reads  +13°,  and  four  hours  previously 
it  read  —7°.  Through  how  many  degrees  and  in  what 
direction  had  the  top  of  the  mercury  changed  meanwhile? 

Present  reading,     +13° 
Previous  reading,   —  7° 

The  change,   +20°,  obtained  by  subtracting  -7°  from  13°. 

2.  Starting  from  a  stair-landing  a  boy  goes  up  17  steps, 
and  drops  his  pencil,  which  rolls  down  to  the  landing,  across 
the  landing,  and  on  down  to  the  6th  step  below  the  landing, 
where  it  stops,  The  steps  are  a  inches  high.  How  far  and 
in  what  direction  must  the  boy  go  to  get  to  the  step  where 
the  pencil  Hes? 

CaUing  upward  +  and  downward  — ,  the  boy- 
arriving  at  —  6a 
starting  from   +17a 

goes   —23a,  meaning  23a  inches  downward. 

In  these  cases  we  have  been  subtracting  signed  numbers. 
Let  us  now  learn  the  general  plan  of  subtracting  such 
numbers. 

35 


36  ELEMENTARY   ALGEBRA 

SUBTRACTING  SIMILAR  TERMS 

56.  The  following  examples  represent  all  cases  in  addition 
with  reference  to  signs  and  relative  values  of  addends : 


5a 

3a 

-5a 

-3a 

-5a 

+5a 

-3a 

3a 

3a 

5a 

-3a 

-5a 

3a 

-3a 

5a 

-5a 

8a 

8a 

-8a 

-8a 

-2a 

2a 

2a 

-2a 

Write  examples  in  subtraction,  using  the  above  sums  as 
minuends  and  one  addend  as  subtrahend,  as  follows: 


8a 

8a 

-8a 

-8a 

-2a 

2a 

2a 

-2a 

3a 

5a 

-3a 

-5a 

3a 

-3a 

5a 

-5a 

5a 

3a 

-5a 

-3a 

-5a 

5a 

-3a 

3a 

By  the  definition  of  subtraction,  the  difference  or  remain- 
der in  each  case  must  be  the  other  addend. 

Show  that  the  correct  result  might  have  been  obtained  in 
each  case  by  changing  the  sign  of  the  subtrahend  and  adding. 

67.  Principle. —  Subtracting  any  number  is  equivalent  to 
adding  a  number  of  equal  absolute  value  but  opposite  quality. 

58.  Rule. — Conceive  the  sign  of  the  subtrahend  changed  from 
-\-  to  —  or  from  —  to  -\-  and  proceed  as  in  addition. 
The  change  of  sign  should  always  be  made  mentally. 

Exercise  15  —  Subtracting  Similar  Terms 

Give  remainders  in  the  following  orally: 

1.  9a  2. 

4a 


e.  5a 

4a 


-4x 
Qx 

3. 
8. 

-35 
-86 

-4b 
-  h 

4.  7n 
2n 

9.  n 
6n 

6. 
10. 

-lie 
-  3c 

3x 
-7x 

-10c 
4c 

SUBTRACTION  OF  MONOMIALS  37 

11.  5x  12.  -46    13.  -  n  14.  9c    15.  -12a 

9a:        56       —5n  c  8a 


16.  66    17.   9n    18.  -8c    19.  a  20.  -  7x 

76       -2n  -  c  3a  -lOx 


21.  2n         22.  -8c    23.  -  a    .   24.  Qx  25.  -146 

7n        7c       —2a       a;       -  66 


26.  8c    27.   3a    28.  -6a;    29.  6    30.    7n 
7c       -4a       -  X  86       -13n 


31.  6a    32.  -7a;    33.  -  6    34.  7n         35.  -lie 
9a        9a:       —46        n  4c 


36.  5x    37.   86    38.  -3n    39.  c  40.  -13s 

6a:       —66       —  n  5c       — lis 


SUBTRACTING  DISSIMILAR  TERMS 

59.  A  man  had  5a6  acres  of  land  and  sold  2xy  acres  of  it. 
How  many  acres  had  he  left?  The  subtraction  of  dissimilar 
terms  is  indicated  by  writing  one  term  after  the  other. 
Thus, 

5a6  3ac  —  2aa: 

2xy  —4xy  —36c 


5ab  —  2xy  3acH-4x!/  36c  — 2aa: 

In  indicating  the  subtraction  of  dissimilar  terms,  the  sub- 
trahend must  be  written  with  its  sign  changed. 


38  ELEMENTARY  ALGEBRA 

Exercise  16  —  Subtracting  Monomials 
Give  remainders  in  the  following  orally : 


..  3a 

2.   -4x 

3.  -4a 

4. 

-5n 

6. 

4x 

b 

-7x 

-2n 

7n 

-  2/ 

:.  7a 

7.  -7a 

8.   -66 

9. 

2n 

10. 

7c 

8a 

-2c 

h 

-3x 

-6c 

11.     a  12.  -3a;  13.   -  a  14.  -3n  15.       5x 

2b              -9x  -Sx  6n  -Sy 

16.     X  17.       3a  18.   -26  19.   -3n  20.       5c 

9a;              -26  76  -4/1  -  c 


21.     a        22.   —   aj  23.   —   a  24.       4n  25.   —   c 

56  -9x  -46  -3n  8c 


26.  3a;  27.       5a  28.   -36  29.   -5n  30.   -2x 

Qx              -2c                 -76  -6a  -5y 

31.  4a  32.         x  33.   -46  34.   -2n  35.   —5c 

9a              — 6i/                    76  -4a  —  c 

36.     a  37.  -7a;  38.  -2a  39.  -4n  40.  -8c 

7a;                  5a;                —26  -9n  c 


41.  3a        42.   -  X  43.       66  44.   -4n  45.       7c 

5a  85/  -86  5c  -9c 


SUBTRACTION  OF  POLYNOMIALS  39 

SUBTRACTION   OF  POLYNOMIALS 

60.  1.  Subtract  7  dollars,  3  quarters,  8  dimes  from  16 
dollars,  7  quarters,  12  dimes. 

Letting  c  be  the  number  of  cents  in  a  dollar,  q  the  number 
of  cents  in  a  quarter,  and  d.  the  number  of  cents  in  a  dime, 
we  write: 

From  lGc+7q+12d 

Take  7c+3g+  Sd 

Difference,    9c+4g+  4d 

2.  From  5ah—4ac-\-Sbc  bushels  of  grain,  4a6  — 6ac+2cc? 
bushels  were  sold.     How  many  bushels  remained? 

Minuend,       5ab  —  4ac-\-Sbc 
Subtrahend,  4a6  — 6ac  -\-2cd 

Difference,      ah  -\-  2ac + 36c  —  2cd 

61.  Rule. —  Write  the  polynomials,  similar  terms  in  a 
column.     Beginning  at  the  left,  subtract  as  with  monomials. 

Subtraction  is  checked  by  determining  whether  the  difference 
between  the  values  of  minuend  and  subtrahend  is  equal  to  the  value 
of  the  remainder.     Observe  the  work  below: 

Work  Check 


5a6-4ac-f36c  =    10-12+18         =16 

.    4ab-6ac  -\-2cd  =     8-18         +24  =  14 

ah-\-2ac+3hc-2cd  =     2+  6+18-24=   2 

The  above  example  in  subtraction  has  been  checked  by 
substituting  1  for  a,  2  for  h,  3  for  c,  and  4  for  d. 

It  is  now  plain  that  subtracting  is  finding  what  number 
must  be  added  to  the  subtrahend  to  give  the  minuend.  Hence, 
another  good  check  on  subtraction  is  to  add  the  subtrahend 
and  difference  and  see  if  the  sum  is  the  minuend. 


40  ELEMENTARY   ALGEBRA 

Exercise  17  —  Subtracting  Poljmomials 
Solve  the  following  and  check  the  first  nine : 

1.  From  8a6-5c+4d-8  subtract  4(i+3a6-12-6c. 

2.  Subtract  5ay — z-\-9ax-{- 14:  from  4axH-6ai/  — 2;+8. 

3.  From  66c-56H-8de+/  subtract  8de+10+56c-56. 

4.  Subtract  4ac+Sbd-2bc-She  from  4ac+26rf-  106c. 

5.  From4:CX-{-7by  —  xy  —  9s\ihtrsiCi7by  —  10-{-3cx-{-xy. 

6.  Subtract  Aax — Axy + lab  from  4aa:  —  2ac  —  Sxy  -h  8a6. 

7.  From  ax  —  7ay-\-dxy  —  2z  subtract  4:xy  —  7ay  —  7-\-ax. 

8.  Subtract  Sab-\-Q  —  7ac  —  ax  —  am  from  4a6  — 6ac  — am. 

9.  From  6am— 4an4-4ar— 7rs  subtract  12H-6am— 12an. 

10.  From  the  sum  of  3a-\-2b  —  Sc+d  and  2d-|-2a— 46  sub- 
tract 36-5+3d+4a+3c. 

11.  From  4:X  —  dy-{-2z  —  u  subtract  the  sum  of  dz-\-2x  —  Q 
—4?/ and  —2z  —  x-\ry  —  2u-\-Q. 

12.  Subtract  the  sum  of  2y-\-2b  —  5x  —  da  and  3a:  — 66+3?/ 
+4a  from  2a-36-2x+4i/. 

13.  From  the  sum  of  3c-2d-5e+2/  and  8e-4d-3/-6c 
subtract  5e  — 5c— /— 3d 

14.  Subtract  26  — 2c +d  — 2a  from  the  sum  of  2a  — 56 -f- 2c 
-2d  and  46+3d-3a-3c. 

16.  From  the  sum  of  Sx-\-2y—z-{-2u+S  and  32-4a:-10 
^by—^u  subtract  2z  —  3  —  bx  —  '3u  —  2y. 

16.  From  5a6+2ac  — 36c  subtract  the  sum  of  26c-|-3ac+6(/, 
4a6  — 26d— 46c,  and  bd—iac  —  ab. 

17.  From   the   sum   of   4:X-{-y  —  2z   and   4iU-\-Sy—7x  —  2z 
subtract  4w+42/  — 52!+5  — 3a;. 


SYMBOLS  OF  AGGREGATION  41 

18.  What  number  must  be  added  to  —4a +66  — 8c  to  give 
0?     To  give  8a+46-4c? 

19.  From  4a6  — 3ac+26c  subtract  the  sum  of  Shc-^-hd—ac, 

Sab  —  Shd — he,  and  bd  —  2ac  —  ab. 

20.  From  the  sum  of  3a  — 2a: +5  and  4:X-\-2y—4:  subtract 
the  sum  of  3a;  — 2aH-3  and  y  —  2x  —  2. 

21.  If  a;  =  5a-36+4c,  y  =  Sa-2b-3c,  z  =  a-\-b+Qc,  find 
the  value  of  x  —  y  —  z. 

22.  What  number  must  be  subtracted  from  2a6  — 3ac— 56c 
to  give  ac  -  56c  +  2a6  -  36d?     To  give  0? 

23.  Subtract  2a-36+4  from  7,  26-3a+3  from  unity,* 
a  — 26+2  from  zero,  and  add  the  three  results. 

SYMBOLS   OF  AGGREGATION 

62.  The  product  8X14  can  be  shown  thus:  8(10+4), 
which  means  8X10+8X4  =  80+32=112. 

This  use  of  the  symbol  (  ),  called  a  parenthesis,  is  of  aid 
in  learning  rapid  mental  calculation,  thus: 

7X25  =  7(20+5)  =  140+35  =  175 
6X49  =  6(40+9)  =240+54  =  294 
9X68  =  9(60+8)  =  540+72  =  612,  etc. 

63.  A  man  walks  north  5  miles  an  hour  for  2  hours  and  then 
south  along  the  same  road  3  miles  an  hour  for  2  hours.  How 
far  is  he  then  from  the  starting-point? 

The  answer  to  this  problem  may  be  written  thus: 

2(5-3)=2X2  =  4 

Show  that  the  perimeter  of  a  rectangle  x  wide 
and  y  long  may  be  written:  2(x+y)  or  2x-\-2y, 
or  x-\-y-\-x-^y  and  that  2{x-\-y)  =2x-\-2y. 

*Unity  means  1. 


Rectangle 


42  ELEMENTARY  ALGEBRA 

64.  In  a  series  of  the  four  operations,  the  multiplications 
and  divisions  are  to  be  performed  first.     Thus, 

8+7X3-6+16-^2+5-5X3-8^2  = 
8+  21   -6+     8     +5-   15-4    =17 

In  such  a  series  the  terms  are  the  parts  separated  by  the  signs  + 
and  — .  The  above  example  contains  seven  terms.  When  such 
expressions  are  to  be  simplified  or  reduced,  each  term  should  be  first 
simplified  or  reduced. 

When  it  is  desired  to  perform  the  operations  of  a  series  in 
any  order  other  than  the  one  mentioned  above,  it  is  necessary 
to  use  some  symbol  of  aggregation. 

65.  The  symbols  of  aggregation  are  the  parenthesis  (  ), 
the  brace  [ } ,  the  bracket  [],  and  the  vinculum . 

These  mean  that  the  operations  indicated  within  them 
are  to  be  performed  before  the  operations  upon  them;  in 
other  words,  that  the  expressions  within  them  are  in  each 
case  to  be  regarded  as  one  number.  Every  part  within  the 
symbol  is  affected  by  the  operation  indicated  upon  the 
symbol.     Observe  the  following: 

18-9-4  =  5  15X12-8    =172 


18-(9-4)  =  13  15X12-8    =60 


216-(24-36-^4)X4- (4+6X3-35-8X4)  =  137 
216-  60  -  19  =137 

Notice  the  use  of  the  parenthesis  in  the  following: 

1.  If  the  smaller  of  two  numbers  is  a;  — 7  and  the  larger 
X— 2,  their  difference  is  (x  — 2)  — (x  — 7). 

2.  If  a  rectangle  is  x+8  in.  long  and  x+3  in.  wide,  the 
area  of  the  rectangle  is  (x+8)(x+3)  square  inches. 

3.  If  J  of  the  distance  between  two  cities  is  x+10  miles, 
the  whole  distance  is  3(x+10)  or  (x+10)3  miles. 


SYMBOLS  OF  AGGREGATION  43 

Exercise  18 
Remove  the  symbols  of  aggregation  and  then  simplify : 


1.  465+67X8- (9X24+144^4-45^5X6) 

2.  764-(245-465-^5)-14X7+789-540-^9 

3.  238-8X9- 108^9+754- (84-58) -47X8 


4.  9X(48+65)  +  128^4-(8Xl2+7-8X8)X7 

66.  Operations  on  Compound  Expressions.  Symbols  of 
aggregation  are  much  used  in  algebra  to  indicate  operations 
on  compound  expressions. 

To  indicate  the  subtraction  or  multiplication  of  a  poly- 
nomial, a  parenthesis  is  necessary. 

Thus,  x{a-\-h)  represents  the  product  of  x  and  a-\-h  and  is 
read  x  times  a +6,  or  a+6  times  x. 

Exercise  19 

1.  Indicate  the  subtraction  of  x  — 5  from  3x+4.  Indicate 
the  product  of  two  binomials. 

2.  If  a  man  has  8a;  sheep  and  sells  2x+35  of  them,  what 
will  denote  the  number  he  has  left? 

3.  What  does   (x-\-5){x  —  2)  represent,  if 
X  represents  the  nuniber  of  feet  on  each  side    "^ 
of  a  square? 

4.  What  does  x{x-^S)  represent,  if  x  stands  for  the  number 
of  rods  on  each  side  of  a  square? 

5.  Represent  in  two  forms  4  times  the  sum  of  any  two 
numbers.     5  times  the  difference  of  any  two  numbers. 

6.  Represent  the  product  of  two  equal  numbers  each  of 
which  is  8  greater  than  x. 

7.  At  85^  a  rod,  express  in  two  ways  the  cost  of  enclosing 
a  rectangular  farm  x  rods  by  y  rods. 


(N 

X 

5 

44  ELEMENTARY  ALGEBRA 

8.  If  X  is  any  positive  integer  greater  than  5,  is  x  — 5 
greater  or  less  than  a:  —  3?     Show  why. 

9.  What  is  the  equation  which  tells  that  the  difference 
between  a;  — 9  and  a:  — 4  is  a? 

10.  If  the  difference  between  x— 12  and  x  — 8  is  n,  what  is 
the  value  of  nl 

11.  If  X  is  any  positive  integer,  when  is  ax  greater  than  x1 
When  is  ax  less  than  x? 

12.  How  many  trees  are  there  in  an  orchard,  if  there  are  20 
more  trees  in  a  row  than  there  are  rows? 

13.  Write  3a  times  the  product  of  two  binomials  divided  by 
the  product  of  a + 6  and  a  —  6. 

14.  Indicate  how  many  acres  there  are  in  a  rectangular  field 
a;  — 8  rd.  wide  and  x+lO  rd.  long. 

15.  What  may  represent  the  product  of  4  numbers,  if  any 
2  of  them  in  order  differ  by  the  same  number? 

16.  Write  an  expression  of  3  terms,  each  term  containing 
one  or  more  compound  factors. 

17.  At  $40  an  acre,  what  is  the  value  of  3  farms  containing 
X,  a:+20,  and  x  — 5  acres,  respectively? 

18.  Represent  the  product  of  two  unequal  numbers,  part 
of  each  number  being  x. 

19.  Wh^t  is  the  area  of  3  equal  rectangles,  the  width  of 
each  being  x  in.  and  the  length  6  in.  greater? 

20.  The  area  of  a  square  x  in.  long  is  the  same  as  that  of 
a  rectangle  x+6  in.  by  x— 4  in.     Express  as  an  equation. 

21.  How  much  does  a  boy  earn,  if  the  number  of  cents  he 
gets  per  day  exceeds  the  number  of  days  he  works  by  20? 


SYMBOLS   OF  AGGREGATION  45 

67.  Extended  Meaning  of  Term.  It  is  necessary  now  to 
enlarge  our  idea  of  a  term,  especially  when  signs  of  aggregation 
are  used.     For  example,  the  expression, 

2a{x-\-y)  -  (3a+5)  -  {2a^\)x-  (a+6)  {a-h) 

contains  only /our  terms. 

In  an  expression  involving  symbols  of  aggregation,  that  part  of  the 
expression  within  the  symbol  of  aggregation  is  to  be  regarded  as  a  term, 
or  as  one  of  the  factors  of  a  term. 

Exercise  20 

1.  Write  3  times  the  sum  of  a  and  h,  diminished  by  5  times 
the  product  of  a,  &,  and  c. 

2.  If  a  rectangle  is  x  inches  long  and  y  inches  wide,  what 
does  2{x-\-y)  represent? 

3.  If  2n  —  1  represents  an  odd  number,  what  will  represent 
the  next  larger  odd  number? 

4.  How  many  square  inches  are  cut  off  in  a  strip  3  inches 
wide  all  around  a  square  of  paper  x  inches  long  and  wide? 

68.  Removing  Symbols  of  Aggregation.  Symbols  of 
aggregation  preceded  by  —  may  be  removed  by  changing 
the  signs  of  the  terms  enclosed.     Thus, 


3a-26-(2a-56+c>= 
3a-26-2a+56-c 

The  reason  for  this  change  is  evident  from  the  principles 
of  subtraction,  as  the  number  enclosed  is  to  be  subtracted. 

Symbols  of  aggregation  that  are  preceded  by  +  are 
removed  without  changing  the  signs  of  the  terms  enclosed. 

The  minus  sign  before  a  symbol  of  aggregation  being  a  sign  of  opera- 
tion, students  should  remember  that  if  the  first  term  of  the  number 
enclosed  has  no  sign  expressed,  it  is  positive. 


46  ELEMENTARY   ALGEBRA 

Exercise  21 

Remove  the  symbols  of  aggregation  in  the  following  and 
express  the  results  in  as  few  terms  as  possible : 

1.  4a-6-(a-26+c)  2.  3x- (~2x-\-Sy)+2y 

3.  3a-(6-c+2a)+6  4.  4x-{-Sy+i-Sx-4y) 


6.  5a-b-4a+h-c  6.  5x- i-2x-4y)-Sy 

7.  2a+h-c-{-{Sa-h)  8.  2x-Sy- {-2x-4y) 


9.  Sa-c-b-4a-b  10.  4n-3x+(-3n-4a;) 

When  x  =  2a-3b+4:c,  y  =  Sa+2b-5c,  z  =  4a-5b-Sc,  find 
the  value  of  each  of  the  following : 

11.  x-{-y-\-z  12.  x+y  —  z  13.   —x  —  y  —  z 

14.  x  —  y  —  z  16.  X  — 2/+2;  16.   — x+i/  — 2 

69.  To  remove  two  or  more  symbols  of  aggregation,  one 
within  another,  begin  with  the  outer  one.* 

3a-{a+2b-a-^p^^n) 

=  3a—  a  —  2b-\-a-{-b  —  c—n 

It  should  be  noted  that  the  —  sign  before  the  b  belongs  to  the  vin- 
culum, not  to  the  6.    The  sign  of  the  6  is  +. 

Removing    the    outer   symbol    changes  the   sign   before 


b—c  to  -\-,  and  these  two  terms  are  brought  down  with  the 
same  signs. 

*Many  teachers  prefer  to  begin  with  the  innermost  symbol  of  aggre- 
gation.    Either  way  becomes  easy  after  a  little  practice. 

It  is  just  about  as  easy  and  it  is  even  quicker,  to  remove  all  symbols 
of  aggregation  at  once  by  beginning  at  the  left  and  bringing  each  suc- 
cessive term  down  with  its  own  or  the  opposite  sign  according  as  there 
is  an  even  or  an  odd  number  of  the  antecedent  minus  signs  affecting 
it.  Any  one  of  the  three  ways  becomes  easy  and  reliable  with  a  little 
practice. 


SYMBOLS  OF  AGGREGATION  47 

Exercise  22 

Remove  the  symbols  of  aggregation  in  the  following  and 
simplify  the  results : 

1.  Qa-{b+5a-{-c)+h  2.  2y-3x- i-4:X-3y) 


3.  2a-(36-a+6-c)  4.  Qx- {-2y+3y-5x) 


5.  4a-{2b-a-\-c-h)  6.  5x- {-2y-4:X-3y) 


7.  5a-{h+a-2h-a)  8.  4n-i-Sx-\-3n-Qx) 


9.  3a-(6-2a+6-c)  10.  7x- (-4?/-3a:+32/) 


11.  4a+(6-a+26-c)  12.  3x+27/-(-2x-4?/) 


13.  2a-(c+6+2a-6)  14.  An-{-2x-{-Sn-\-2y) 


15.  36-(a+36+a+c)  16.  2y-i-2x-3x-Sy) 

70.  It  follows,  that  in  order  to  enclose  two  or  more  terms 
of  a  polynomial  in  a  symbol  of  aggregation  preceded  by  the 
sign  — ,  we  must  change  the  signs  of  the  terms  enclosed. 
Thus, 

ab  —  ac-{-hc  —  cd  =  ab—  {ac—bc-{-cd) 

Exercise  23 

Enclose  the  last  three  terms  of  each  of  these  polynomials 
in  a  parenthesis  preceded  by  a  minus  sign: 

1.  ac—ax-\-ab-{-bx  2.  2x-{-2y—xy  —  xz-{-yz 

3.  ab-{-bc  —  ac-^ax  4.  ax  —  ay  —  2x-\-xy  —  2y 

5.  ax  — bx  — be— by  6.  3a+26+aa;  — a6+6c 

7.  an-\-ab-\-ac—bc  8.  2a  —  ab  —  ax-\-bc  —  2c 

9.  ac  —  ax  —  bc-\-bx  10.  bc-{-2a-\-ac-\-2x  —  ac 


48  ELEMENTARY  ALGEBRA 

ADDITION   OF  TERMS  PARTLY  SIMILAR 

71.  Terms  that  are  partly  similar,  i.e.,  similar  as  to  part 
of  the  letters  only,  may  be  united  into  one  term  with  a 
polynomial  coefficient.     Thus, 

ay  ax  an 

by  X  —2n 

{a+h)y  {a-{-l)x  {a-2)n 

72.  Rule. —  Write  the  dissimilar  parts  in  a  parenthesis  as 
the  polynomial  coefficient  of  the  similar  part. 

The  above  answers  are  read:  "a  plus  h,  times  y^';  *'a  plus 
1,  times  a:";  and  ^'a  minus  2,  times  n, "  a  slight  pause  in  the 
reading  occurring  where  the  last  curve  of  the  parenthesis 
stands.  . 

Exercise  24 

Read  the  sums  of  the  following : 
1.  ax  2.       by  3.       an  4.  ax  5.    —by 

bx  —  y  —  3n  x  cy 


ax  8.   —an  9.  ab 

4:X  n  2b 


6. 

y 
bi 

11. 

Sx 

ax 

16. 

Sx 

ax 

X 

21. 

ax 

X 

ax 

12.       an         13.  —by  14.  ar 

—  en  y  cr 


17.   -Ay  18. 

xy 

-  y 


10. 

—  n 

an 

15. 

-  5x 

—  ax 

22.       ay         23. 

-  y 
-2y 


4:X 

19.  bx 

20. 

xy 

—  cx 

X 

-  y 

X 

bx 
24.  ax 

25. 

xy 

Sx 

-xy 

-2x 

X 

y 

—  nx 

bx 

-xy 

SUBTRACTION   OF   TERMS   PARTLY   SIMILAR         49 
SUBTRACTION   OF  TERMS   PARTLY   SIMILAR 

73.  Terms  partly  similar,  i.e.,  similar  as  to  part  of  the 
literal  factors,  may  be  subtracted  by  indicating  the  sub- 
traction of  the  dissimilar  parts.     Thus, 

ax  hy  n 

hx  —cy  an 

(a  —  b)x  (b-{-c)y  •  (l  —  a)n 

74.  Rule. —  Write  the  indicated  subtraction  of  the  dissimilar 
parts  in  a  parenthesis  as  a  polynomial  coefficient. 

Observe  that  the  sign  of  the  dissimilar  part  in  the  subtra- 
hend is  changed  from  +  to  — ,  or  from  —  to  +. 

Exercise  26 

Subtract  and  read  the  results  of  the  following : 
1.    ay  2.—bx  3.      4a  4.    ax  5. —an 

cy  —ax  —46  x  '    2n 


6. 

ax 

7. 

n 

2x 

—an 

11. 

b 

12. 

-  y 

nb 

-xy 

16. 

xy 

17. 

—    X 

ax 

ax 

21. 

bx 

22. 

y 

ax 

-by 

26. 

y 

27. 

—  an 

by 

nx 

8. 

—  ax 

xy 

13. 

-2x 

2y 

18. 

nx 

-xy 

23. 

-3a 

-3x 

28. 

by 

-41/ 

9. 

ac 

10. 

-2c 

ex 

—  ac 

14. 

nx 

16. 

ac 

an 

-be 

19. 

c 
ac 

20. 

-ay 
by 

24. 

3?/ 
by 

26. 

—  an 

—  n 

29. 

be 

30. 

-by 

cy 

y 

CHAPTER  V 

GRAPHING  FUNCTIONS.    SOLVING  EQUATIONS 
IN  ONE  UNKNOWN  GRAPHICALLY 

GRAPHING  FUNCTIONS 

75.  Algebraic  Numbers,  or  Functions.  For  the  present  it 
is  convenient  to  call  a  number  expressed  by  the  aid  of  one  or 
more  letters  an  algebraic  number  or  a  function  of  the 
numbers  denoted  by  the  letters.  Thus,  2a: +3,  n^  —  2n  —  S, 
a-\-h,  x  —  y,  etc.,  are  algebraic  numbers  or  functions. 

The  n^,  in  n^  — 2n— 8,  means  nXn  and  is  read  n-square,  just  as  5^ 
means  5X5  and  is  read  5-square. 

With  every  algebraic  number  or  function,  such  as  3a: +5 
(or  n^  —  2n  —  S),  two  numbers  must  be  thought  about,  viz. :  the 
algebraic  number  or  function  itself  and  the  number  x,  or  n, 
that  it  depends  on  for  its  value.  The  number  n^  —  2n  —  8  tells 
us  to  form  a  compound  number  by  squaring  some  simple 
number  (n),  subtracting  twice  the  simple  number,  and  then 
subtracting  8.  The  two  numbers  to  be  thought  about  are 
the  value  of  n^  — 2n  — 8  itself,  and  the  value  of  n,  and  so  for 
other  compound  numbers.  The  number  x,  n,  t,  or  y,  in  terms 
♦of  which  the  compound  number  (the  function)  is  expressed, 
may  be  called  the  independent  number. 

In  other  words,  the  value  of  3a; +5  depends  on  what  x  is, 
and  the  value  of  71^  —  271  — 8  depends  on  the  value  of  n.  The 
x  and  the  n  are  the  independent  numbers. 

For  the  reasons  just  stated,  a  number  expressed  in  terms  of 
X,  such  as  3a:  4- 5,  is  called  a  function  of  x,  and  is  written  f(x) 
and  read :  function  of  x. 

Similarly,  n^  — 2n  — 8  or  any  other  number  expressed  in 
terms  of  n,  may  be  denoted  by  f{n)  and  read :  function  of  n. 

50 


GRAPHING   FUNCTIONS  51 

A  function  is  a  number  that  depends  on  some  other  number 
for  its  valu£. 

An  algebraic  function  is  a  number  whose  dependence  on 
another  number  is  expressed  in  algebraic  symbols,  as  3a; +5, 
n^  —  2n  —  S,  a-\-b,  x  —  y,  etc. 

In  this  book  the  word  ' 'function"  means  algebraic  function. 

A  function  that  depends  on  two  other  numbers,  as  a-\-b, 
is  denoted  by  /(a,  b)  and  read:  function  of  a  and  b.  Thus, 
also  a;  —  ^  is  denoted  by  f{x,  y)  and  read :  function  of  x  and  y. 

The  parenthesis,  (  ),  in  the  function  symbol  does  not  mean  multipli- 
cation, but  is  a  part  of  the  symbol. 

If  the  letter  within  the  (  )  is  replaced  by  a  positive  or 
negative  arithmetical  number,  as  in  /(  — 2),  the  meaning  is 
that  the  number,  —2,  is  to  be  substituted  for  the  letter  in 
the  function.     Thus, 

If/(x)=3x+5,  then /(-2)  =3- -2+5= -6+5= -1,  and 
if/(n)=n2-2n-8,  then /(5)  =52-2-5-8  =  25-10-8  =  7. 

Find/(3)if/(x)=8a:-3. 

Find/(-4)  if /(n)=3n+15. 

76.  Two   very   important   problems   of   algebra   are: 

I.  Knowing  the  value  of  the  independent  number,  to  find  the 
value  of  the  function;  and 

II.  Knowing  the  value  of  the  function,  to  find  the  value  of 
the  independent  number. 

77.  We  already  know  how  to  solve  Problem  I. 

For  example,  to  find  the  value  of  3a; +5  for  a;  =  4,  we  have 
only  to  substitute  4  for  x  in  3a; +5,  thus  3X4+5.  Reducing, 
we  find  3x+5  =  17  for  a;  =  4,  and  so  also  for  any  other  value 
of  X.  To  find  the  value  of  n^- 2n  — 8  for  some  value  of  n, 
as  5,  we  substitute  5  for  n,  thus  5^-2X5-8  =  7,  to  see  that 
71^  —  2n  —  8  =  7  for  n  =  5,  and  so  for  other  values  of  n. 


52 


ELEMENTARY  ALGEBRA 


Thus  we  know  that  to  solve  the  first  of  the  above  problems,  we 
have  only  to  substitute  the  value  of  the  independent  number  in  the 
function  and  to  simplify. 

78.  The  second  problem  occurs  very  frequently  in  algebra, 
viz. :  To  find  the  value  of  the  independent  number  when  the 
valy£  of  the  function  is  known.  This  is  Problem  II  above 
and  it  is  the  converse  of  Problem  I.  For  example,  it  is  often 
necessary  to  solve  such  problems  as : 

Given  3x+5  =  8,  to  find  the  value  of  x,  or 
Given  n^  —  2n  —  S  =  7,  to  find  the  value  of  n. 

Such  expressions  as  3a:+5  =  8  and  n^  — 2n  — 8  =  7  are  equa- 
tions, and  to  solve  them  means  to  find  what  value  or  values, 
of  X  or  of  n  will  make  3a: +5  equal  to  8,  orn^  — 2n  — 8  equal  to 
7.  Consequently,  to  solve  the  second  problem  stated  above 
(§  76,  II),  requires  a  knowledge  of  the  ways  of  solving  equa- 
tions. We  shall  first  show  by  means  of  pictures  what  it 
means  to  solve  equations. 

Let  it  be  kept  in  mind  that  alge- 
braic equations  are  made  up  of 
algebraic  numbers. 

79.  Dependence  of  an  Algebraic 
Number,  or  Function.  Let  us  first 
try  to  understand  the  relation  that 
exists  between  x  and  3x+5. 

Draw  a  vertical  and  a  horizontal 
algebraic  scale  (YY'  and  XX^so  that 
they  shall  be  at  right  angles,  with 
their  0-points  together,  as  shown  in 
the  figure.  This  is  quickly  done 
with  cross-lined  paper.  Pupils 
should  have  some  pages  of  cross- 
lined  paper  in  their  note-books.  Graph  of  3a: +5 


h           '■'     ^-f 

\1    zL 

iL         4  n/^ 

it^ 

??/ 

\lWf 

lOH 

^y~ 

/I 

I_ 

-^  *^ 

3x    ' 

jf^  ^ 

-6-5-4-3-2/           ' 

V -'. 

/C?      ' 

i    - 

,Gr  -4 

j"   - 

AT         Te 

-p  -J 

GRAPHING  FUNCTIONS  53 

Now  proceed  thus: 

Assume  x=l,      2,      3,      4,    0,-1,-2,-3,-4, 

and  calculate  3a:+5  =  8,    11,    14,    17,    5, +2,  -1,  -4, -7. 

These  rows  of  numbers  mean  that  1  and  8  go  together,  as 
also  2  and  11,  3  and  14,  and  so  on  to  —4  and  —7.  They 
are  number-pairs,  that  are  paired  through  3x4-5,  and  are 
usually  written:  (1,  8),  (2,  11),  (3, -14),  (4,  17),  (0,  5), 
(-1,  +2),  (-2,  -1),  (-3,  -4),  and  (-4,  -7).  The  x- 
value  is  always  written  first. 

Now  picture  the  number-pair  (1,  8)*  by  starting  from  the 
0-point,  measuring  1  unit-space  to  the  right  and  then  8 
units  upward  vertically,  and  marking  the  point  reached,  as  A . 
This  point.  A,  pictures  the  number-pair   (1,  8),  for  it  is 

I  unit  from  the  vertical  scal§.  and  8  units  from  the  horizontal 
scale. 

Then  picture  the  number-pair  (2,  11)  by  starting  from  the 
0-point,  measuring  2  units  horizontally  to  the  right,  and  then 

II  units  vertically  upward  to  B.     The  point  B  pictures  the 
number-pair  (2,  11). 

Similarly  picture  the  number-pairs  (3,  14),  (4,  17),  (0,  5), 
(  — 1,  +2),  measuring  minus  values  of  x  from  0  toward  the 
left,  (  —  2,  —1),  measuring  minus  values  of  3x+5  downward 
from  the  horizontal,  (  —  3,  —4),  and  (  —  4,  —7)  as  at  C,  D, 
E,  F,  G,  H,  and  /. 

If  your  measuring  and  your  work  are  correct,  and  you 
stretch  a  string  tightly  just  over  the  points,  you  will  find 
them  to  lie  on  a  straight  line. 

If  you  do  so  find  them,  draw  the  straight  line  through 
them. 

*Remember  that  if  no  signs  are  written  before  numbers,  the  plus- 
sign  (+)  is  understood.  Thus,  (1,  8)  means  (-+-1,  +8),  and  (2,  —5) 
means  (+2,   —5),  etc. 


54 


ELEMENTARY   ALGEBRA 


If  you  should  substitute  any 
whole  or  fractional  positive  or  nega- 
tive value  in  3a: +5  for  x  and  locate 
the  point-picture  of  the  resulting 
number-pair,  you  would  always  find 
that  the  point  falls  on  this  same  line. 


Tryx  =  i    li  - 


2h,  etc. 


The  conclusion  is  that  3x4-5  con- 
nects   numbers   into   number-pairs, 
whose  picturing  points  all  lie  along  ^ 
the  same  straight  line. 

Any  number  of  pairs  of  values 
are  given  by  3a: +5. 

What  we  have  been  doing  in  this 
section  is  called  graphing  the  func- 
tion 3x+5. 


1 1 

-"i 

OO 

W 

' 

©* 

1 

r 

- 

K] 

i- 

™- 

.J 

r 

^1 

- 

J 

- 

- 

V 

P- 

- 

V 

n  ^ 

Yr 

V 

\ 

\ 

y 

r 

fj 

{ 

1^ 

\ 

\ 

L 

G 

V 

A^l 

1 

-yATT 

80.  Picturing  n2-2n- 8. 

n^-2n-S. 


Graph  of  n2-2n-8 
Let  us  now  make  a  picture  of 


Assume            n  = 

1, 

2, 

3, 

4,      5,        6, 

then  calculate 

n2-2n-8  = 

-9, 

-8, 

-5, 

0,  +7,  +16 

0,  -1,  -2,  -3,     -4, 

■8,-5,      0,  -f7, +16. 

The  number  pairs  are  here  (1,  —9),  (2,  —8),  (3,  —5), 
(4,0),  (5,7),  (6,  16),  (0,  -8),  (-1,  -5),  (-2,0),  (-3,  +7), 
and  (—4,  +16),  the  n-value  being  the  first  number  of  each 
pair. 

Using  again  a  pair  of  perpendicular  algebraic  scales  on 
cross-lined  paper,  picture  the  number-pairs  as  in  the  figure. 

Draw  carefully  freehand  a  smooth  curve  through  points 
A,  B,  C,  and  so  on  to  F  and  then  to  L,  as  shown. 

In  this  case  the  number-pairs  lie  along  a  curve,  called  a 
parabola.     The  parabola  iS  an  open  curve. 


SOLVING  EQUATIONS   GRAPHICALLY  55 

Any  value  you  might  take  for  n,"  substituted  in  n^  — 2n  — 8, 
would  give  a  number-pair  whose  point-picture  would  lie  on 
this  same  curve. 

Try  n  =  |,  n=  —\,  n  =  l\,  n=  —l\,  etc. 

The  function  n^  — 2n  — 8,  is  then  a  number-law  which 
pictures  into  a  parabola. 

What  we  have  just  been  doing  in  this  section  is  called 
graphing  f(n)=n2  — 2n  — 8. 

81.  To  make  pictures  of  functions  we  merely  assume 
values  for  x,  or  n,  etc.,  substitute  the  assumed  values  in  the 
functions  (3x+5  or  n^  — 2n  — 8),  and  calculate  the  second 
numbers  of  the  number-pairs.  It  then  remains  to  picture 
the  number-pairs  on  a  pair  of  perpendicular  algebraic  scales, 
as  above. 

Any  number  of  number-pairs  are  given  by  either  3x+5 
or  n^  — 2n— 8,  or  by  any  other  function. 

Every  such  function  has  some  straight  or  curved  line- 
picture.  The  particular  number-pairs  given  by  any  function 
always  picture  into  points  all  of  which  lie  on  the  same  straight 
or  curved  line.  Hence,  every  function  has  its  own  particu- 
lar line-picture. 

The  rising  and  falling  of  the  line  or  curve  picture  the 
changes  in  the  function  that  are  produced  by  changing  the 
independent  number,  as  x  or  n. 

SOLVING    EQUATIONS   IN    ONE    UNKNOWN    GRAPHICALLY 

82.  Solving  3xH-5  =  8,  Graphically.  Suppose  now  that  we 
were  required  to  solve  the  equation  3x+5  =  8. 

We  would  calculate  some  number-pairs  of  So; +5,  locate 
the  picturing  points  (see  figure  in  §  79),  and  draw  through 
the  points  the  straight  line. 


56  ELEMENTARY  ALGEBRA 

So  soon  as  we  know  the  line-picture  to  be  a  straight  line, 
two  rather  widely  separated  points  are  sufficient  to  give  the 
line-picture. 

Since  we  want  to  j&nd  the  value  of  x  that  makes 

3x+5  =  8, 

we  measure  8  units  up  on  the  vertical  scale,  and  draw  a 
horizontal  out  until  it  crosses  the  line  of  3a: +5.  The  length 
of  this  line,  or  its  equal  measured  along  the  horizontal  scale, 
is  the  required  value  of  x.  The  length  is  1,  and  as  it  extends 
to  the  right,  x=-\-l.  This  horizontal  is  called  the  graphical 
solution  of  3x+5  =  8. 

Notice  that  while  any  number  of  number-pairs  are  given 
by  3a;+5,  only  one  of  these  number-pairs  will  make  3a;+5  =  8. 

83.  Solving  n2-2n-8  =  7,  Graphically.  Similarly,  let  it 
be  required  to  solve  n^  — 2n  — 8  =  7,  graphically. 

Calculate  some  number-pairs  by  substituting  values  of  n 
as  in  §  80,  and  draw  the  parabola-picture,  freehand,  as  in 
§  80. 

Since  we  are  seeking  the  value  of  n  that  makes 

n2-2n-8  =  7, 

we  draw  a  horizontal  through  a  point  7  units  up  on  the 
vertical  scale,  and  prolong  the  horizontal  both  ways  until  it 
crosses  the  parabola.  The  line  is  KE  in  the  figure  of  §  80. 
It  will  cut  the  parabola  in  two  points.  The  lengths  of  the 
parts  of  the  horizontal  between  the  vertical  scale  and  the 
curve  are  the  two  values  of  n  that  will  make 

n2-2n-8  =  7. 

The  two  values  are  n=  +5,  and  n=  —  3. 

Substitute  each  of  the  two  values  in  n^— 2n— 8  and  see  if 
they  make  it  equal  7.  This  shows  that  there  are  two  values 
that  will  give  the  one  value  7  for  the  algebraic  number, 
n2-2n-8. 


SOLVING  EQUATIONS  GRAPHICALLY       57 

Notice  then  that  while  any  number  of  number-pairs  are 
given  by  n^  —  2n  —  8,  only  two  of  these  pairs  make 

n2-2n-8  =  7. 

This  means  there  are  only  two  points  on  the  graph  of 
n^  —  2n  —  8  where  n^  —  2n  —  8  =  7.  They  are  the  points  K  and 
E  in  the  figure  of  §  80. 

84,  We  have  now  shown  how  to  make  pictures  of  number- 
laws  such  as  3rc+5  and  n^  — 2n  — 8,  and  have  also  shown  how 
to  solve  graphically  such  equations  as  3a;+5  =  8  and 
n^  — 2n  — 8  =  7.  For  any  other  algebraic  numbers  or  equa- 
tions that  contain  only  one  letter,  the  method  is  the  same. 


Exercise  26 

Draw  the  Hne-pictures  of  the  following  functions  of  x : 
1.  2x+5                       2.  x+5  3.  a:+3 

4.  2a;+3  6.  3a;+2  6.  3a;-f  1 

7.  3a;- 1  8.  2x-l  9.  a:^-}- 8a; -1-12 

10.  a;2-3a;-10  11.  x^-2x-3  12.  x^-l 

13.  x^-Qx-{-S  14.  X"-6x+5  15.  x~-4x 

Exercise  27 

Solve  the  following  equations  graphically: 
1.  2x+5  =  7  2.  a;+5  =  9  3.^:4-3  =  5 

4.  2a;+3=-l  5.  3a;-j-2  =  8  6.  3x4-1  =  7 

7.  3a;-l  =  5  8.  2a;-l=-5         9.  a;2-f8x+12  =  21 

10.  x2_3a;_io  =  0       11.  x^-2x-3  =  5  12.  rc2_i=8 


58  ELEMENTARY  ALGEBRA 

SUMMARY 

85.  The  work  of  this  chapter  has  taught  the  following 
facts: 

1.  Algebraic  numbers,  or  functions,  require  us  to  keep  in 
mind  two  numbers,  the  function  itself  and  also  some  other 
number,  as  x  or  n,  that  it  depends  on  for  its  value. 

2.  An  algebraic  number  or  function  is  a  shorthand  descrip- 
tion of  the  way  to  calculate  its  own  value. 

3.  Algebraic  numbers  associate  numbers  into  number- 
pairs. 

4.  The  point-pictures  of  the  number-pairs  of  an  algebraic 
number  give  the  line-pictures  of  the  algebraic  numbers, 
called  the  graphs  of  the  algebraic  numbers. 

5.  To  find  the  value  of  an  algebraic  function  when  the  value 
of  the  number  it  depends  on  is  given,  we  substitute  the 
given  value  and  simplify. 

6.  To  find  the  value  of  the  independent  number  when  the 
algebraic  function  is  given  equal  to  a  number,  we  must 
solve  an  equation. 

7.  An  equation  is  only  a  shorthand  way  of  saying  a 
function  is  to  have  a  certain  value. 

8.  While  an  algebraic  function  may  furnish  a  great  number 
of  number-pairs,  usually  only  one  or  a  few  of  these  pairs 
furnish  a  solution  of  the  equation  which  gives  the  algebraic 
function  a  particular  value. 

Although  the  graphical  solutions  of  equations  make  the 
meaning  of  solutions  clear  and  comprehensible,  even  in 
minute  details,  still  they  are  more  tedious  and  cumbersome 
than  the  algebraic  solutions.  When  it  is  only  the  results  of 
solutions  that  are  wanted,  and  after  it  is  learned  that  alge- 
braic solutions  are  shorter  and  easier  ways  of  reaching  these 
results,  we  shall  use  algebraic  solutions. 

Algebraic  solutions  are  treated  in  the  next  chapter. 


CHAPTER  VI 
EQUATIONS.     GENERAL  REVIEW 

EQUATIONS 

86.  The  equation  is  the  backbone  of  algebra.  Its  value 
consists  in  its  power  as  a  tool  for  solving  problems.  Other 
algebraic  topics  are  needed  to  give  insight  into  and  power  over 
the  equation.  Algebraic  skill  means  and  always  has  meant 
nearly  the  same  as  skill  in  using  the  equation.  In  mathe- 
matical history  the  evolution  of  the  equation  means  the 
evolution  of  algebra. 

The  earliest  algebraists  were  the  Egyptians.  Thirty-five 
hundred  years  ago  they  said  such  things  as,  '^A  quantity, 
its  half  and  third  make  19.  Find  the  quantity."  They  used 
no  s5mibols  or  abbreviations,  but  the  language  of  words  only. 

About  sixteen  hundred  years  ago  Diophantus,  a  Greek 
mathematician,  wrote  down  the  initial  letters  of  the  verbal 
sentence  as  his  equation.     It  was  simply  a  shortened  sentence. 

A  thousand  years  later  calculators  wrote  down  rules  for 
calculating  in  symbols,  much  as  a  postal  clerk  of  our  day 
might  write  down  rules  for  calculating  the  postage  on  parcels 
for  various  zones.  For  example,  if  for  zone  3  the  postal  rule 
is  ''6j^  for  the  first  pound  or  fraction  and  2f^  for  each  additional 
pound  in  the  weight  of  the  package,"  the  postal  clerk 
might  write  2x+4,  in  which  x  is  the  weight  in  pounds,  as  a 
short  form  of  the  rule.  On  weighing  the  package  he  might 
do  as  2x4-4  says,  i.e.,  double  the  number  of  pounds  and  add  4 
to  get  the  number  of  cents  to  charge  as  postage. 

Now  if  at  the  other  end  of  the  route  the  persons  receiving 
the  package  had  no  scales  and  desired  to  know  the  weight 
of  the  package,  knowing  the  postage  to  be  12^,  they  might 


60  ELEMENTARY  ALGEBRA 

write  down  2x+4  =  12,  and  find  what  x  is,  if  they  could 
solve  the  equation. 

Again,  if  a  man  starts  5  miles  from  his  home  and  walks 
away  from  it  x  miles  an  hour  for  2  hours,  the  rule  for  finding 
his  distance  from  home  would  be  2x+5.  Suppose  he  did 
not  know  his  rate  but  did  know  how  far  he  was  from  home, 
say  13  miles.  To  find  his  rate  he  might  write  2x-\-5  =  13 
and,  if  he  knew  how  to  solve  the  equation,  he  could  find  his 
rate,  x,  of  walking. 

At  a  later  date  men  came  to  regard  such  forms  as  2x+4 
and  2a: +5,  not  as  shortened  rules,  but  as  the  results  of  fol- 
lowing the  rules,  i.e.,  as  numbers.  Then  they  began  to 
apply  the  laws  of  number  to  them,  that  is  they  began  learning 
how  to  add,  subtract,  multiply,  and  divide  them,  and  alge- 
bra was  a  reality. 

87.  Equations  expressed  partly  or  wholly  in  letters  are 
either  identities  or  conditional  equations. 

88.  An  identity  is  an  equation  with  like  members,  or  mem- 
bers which  may  be  reduced  to  the  same  form. 

89.  The  sign  of  identity  is  = .  It  is  read,  is  identical  with, 
or  is  identically  equal  to,  or  simply  is. 

The  sign  of  equality  may  also  be  used  in  an  identity  when  there  is  no 
need  to  distinguish  the  nature  of  the  equality. 

Thus,  5a-\-4a-\-2a  =  8a+3a,  and  ax-\-c  =  c-\-ax  are  identities,  and  it  is 
evident  that  they  are  true  for  any  value  of  each  letter  in  them. 

90.  Substitution  is  the  process  of  putting  one  number 
symbol  into  an  expression  in  place  of  another  which  has  the 
same  value. 

91.  Satisfying  an  Equation.  An  equation  is  said  to  be 
satisfied  by  any  number  which,  when  substituted  in  place  of 
the  unknown  number,  reduces  the  equation  to  an  identity. 

The  equation,  5x-\-Sx  =  72,  is  satisfied  by  x  =  9,  for  the  substitution 
of  9  for  X  gives  the  identity,  45+27  =  72. 


EQUATIONS  61 

In  the  equation,  5x  —  x  =  3d,  since  5x—x  is  4x,  5x  —  x  and 
36  are  equal  only  when  x  represents  9. 

Thus  5x— x  =  36  is  a  conditional  equation,  because  it  is 
true  only  for  a  particular  value  of  the  letter  in  it. 

Any  equation  may  be  reduced  to  an  identity  by  putting  the 
value  of  each  letter  in  place  of  that  letter. 

The  equations  used  in  solving  problems  are  equations  of  condition. 
The  conditions  of  the  problem,  which  are  expressed  in  language,  are 
translated  into  the  language  of  an  equation. 

92.  A  root  of  an  equation  is  any  value  of  the  unknown 
number  that  satisfies  the  equation. 

In  solving  equations,  we  shall  often  get  an  equation  one  or 
both  members  of  which  are  negative,  such  as, 

-3a:=-12  -5i/  =  35 

It  will  be  explained  later  that  in  such  cases  the  signs  of 
both  members  may  be  changed. 

When  -3x=-12,  3x=12,  and  x=4.  When  -5?/  =  35, 
5?/=  -35,  and  y=  -7.     When  -4s  =  27,  s=  -6f . 

Exercise  28 — Oral  Work 
Solve  the  following  equations: 

1.  -4a;=-15  2.  -3^  =  18  3.  17  =  2a;  4.  42= -51/ 
5.  -5x=-24  6.  -7?/  =  49  7.  20  =  3a;  8.  15= -6?/ 
9.  -8a;=-44     10.   -5y  =  Q0     11.  30  =  4x     12.  62= -3?/ 

93.  In  solving  problems,  it  may  be  necessary  to  multiply 
or  divide  a  term  or  a  binomial  by  some  number. 

To  multiply  3a  by  2,  multiply  the  coefficient  by  2;  to  divide 
8a;  by  2,  divide  the  coefficient  by  2.     Thus, 

4aX2  =  8a  9a-^3  =  3a  nX7  =  7n  Qx-7-Q  =  x 

To  multiply  2a +36  by  2,  multiply  both  terms  by  2;  to 
divide  Qx—12y  by  3,  divide  both  terms  by  3. 


62  ELEMENTARY  ALGEBRA 

Exercise  29 

Perform    the    indicated    operations    and    answer    the 
questions  in  the  following: 

1.  (3a+66)X2  {Sn-10)XS  (8a; -20?/) ^4 

2.  If  x+S  is  the  present  age  of  a  man,  how  old  is  another 
man  who  is  twice  as  old? 

3.  (5x-12)X3  (4a-86)-^4  (6x+120)^6 

4.  If  Tom  has  x  dollars  and  Frank  3x  — 20,  how  many  has 
Fred  who  has  half  as  many  as  both  the  others? 

5.  (8x-92/)X4  (2n+15)X4  (5a-356)^5 

6.  If  X  is  one  number  and  2a;— 10  another,  what  is  a  third 
number  which  is  twice  the  sum  of  the  other  two? 

94.  In  the  statement  of  many  problems,  one  or  both  mem- 
bers may  contain  a  known  and  an  unknown  number.    Thus, 

7a;-4  =  8+5a; 

Before  solving,  it  is  necessary  to  have  all  unknown  numbers 
in  one  member  and  all  known  numbers  in  the  other  member. 

If  by  the  addition  axiom,  §  15,  we  add  +4  and  —5x  to 
both  members  of  the  equation  without  uniting  similar  terms, 
we  have 

7a;-5a;  =  8-f4 

The  same  result  might  have  been  obtained  by  subtracting 
+5x  and  —4  from  both  members  of  the  equation. 

95.  This  process  of  changing  a  term  from  one  member 
of  an  equation  to  the  other  without  destroying  the  equality 
is  called  transposition. 

To  avoid  mechanical  work  and  to  impress  upon  themselves  what 
axiom  is  involved  in  this  change,  students  should  always  explain  the 
work  by  telling  what  they  add  to  or  subtract  from  both  members. 


EQUATIONS  63 

Exercise  30 

In  like  manner  solve  and  check  the  following  equations, 
applying  the  addition  and  the  subtraction  axioms  alternately: 

1.  5a;-32  =  3x-16  2.  14-4n  =  n+32-8/i 

3.  13-6s  =  25-9s  4.  96+12  =  66+40-6 

5.  8i/+14=4!/+74  6.  15-3x  =  x+75-9x 

7.  9n-19  =  44+2n  8.  3s-s-18  =  36-8s 

9.  32-2a;  =  72-6a;  10.  7a+6-15  =  79-4a 

11.  66+16  =  36+26  12.  10+9n  =  88+2n-8 

13.  34-56  =  49-86  14.  6a:-14  =  56-2a:+2 

16.  9s-13  =  4s+27  16.  16+4n+7  =  3n+30 

17.  23-3a:  =  71-7a;  18.  4a-15-a  =  35-2a 

Exercise  31  —  Oral  Practice 
Do  this  entire  list  of  14  exercises  in  26  minutes. 

1.  A  has  X  sheep,  and  B  has  y.     How  many  would  C  have, 
if  he  had  twice  as  many  as  A  and  B? 

2.  Indicate  by  use  of  parentheses  the  product  of  the  sum 
and  difference  of  any  two  numbers,  as  m  and  n. 

3.  If  there  are  x  hundreds,  y  tens,  and  z  units  in  a  number, 
what  will  represent  the  number? 

4.  What  will  represent  the  sum  of  four  consecutive  odd 
numbers  of  which  n  is  the  largest? 

5.  How  many  square  feet  are  there  in  the  walls  of  a  room 
X  feet  square  and  n  feet  high? 

6.  The  sum  of  the  ages  of  4  men  is  lOx  years.     What  was 
the  sum  of  their  ages  12  years  ago? 


64  ELEMENTARY  ALGEBRA 

7.  If  n  represents  an  integer,  does  2n-\-2  represent  an 
even  or  an  odd  number?     Show  why. 

8.  From  x  dollars  a  man  paid  two  debts,  one  of  a  dollars 
and  the  other  of  h  dollars.     How  much  did  he  have  left? 

9.  A  paid  x  dollars  for  a  harness  and  Ax  dollars  for  a 
horse.     Represent  the  cost  of  both. 

10.  If  one  part  of  x  is  16,  what  is  the  other  part?     If  one 
part  of  y  is  45,  what  is  the  other  part? 

11.  A  boy  bought  x  oranges  at  m  cents  apiece  and  sold  them 
at  n  cents  apiece.     If  he  lost,  what  was  his  loss? 

12.  The  difference  between  two  numbers  is  25,  and  the 
smaller  number  is  s.     What  is  the  larger  number? 

13.  Represent  the  number  of  acres  in  a  rectangle  of  land 
x  rods  long  and  x  — 5  rods  wide. 

14.  A  house  cost  n  dollars,  a  farm  5n  dollars,  and  a  store* 
4n  dollars.     Express  in  two  ways  the  cost  of  all. 

Exercise  32  —  Review  Problems  and  Equations 
Solve  and  check  the  following  problems  and  equations: 

1.  The  sum  of  two  numbers  is  128,  and  their  difference  is 
34.     Find  the  larger  number. 

2.  7a;-13  =  x+12+5  3.  6s+17  =  45-2s+8 

4.  Divide  the  number  184  into  two  parts  so  that  the  greater 
shall  exceed  the  less  by  48. 

6.  9n-80  =  26-n-f4  6.  3?/+12  =  16-5?/+4 

7.  The  sum  of  two  numbers  is  270,  and  their  difference  is  4 
times  the  smaller.     Find  the  numbers. 

8.  18-f3a:  =  40-a:+7  9.  76-50  =  23-26-1 


EQUATIONS  65 

10.  A  and  B  own  a  farm  worth  $13,100.  A  has  3  times  as 
large  a  share  as  B.     How  much  is  B's  share? 

11.  4n-15+n  =  5-5n  12.  60-3s  =  6s-8s+7 

13.  One  automobile  ran  3  times  as  fast  as  a  second  and 
6  miles  an  hour  faster  than  a  third.  The  sum  of  their  rates 
was  120.     Find  the  rate  of  the  third. 

14.  16+5x-38  =  7-a;  15.  8a+30  =  35+7a-3 

16.  Three  times  a  number  diminished  by  57,  is  equal  to 
twice  the  number  increased  by  68.     Find  the  number. 

17.  82/-40  =  50-2/+6  18.  9n-15  =  37+2n-h4 

19.  A  horse  and  carriage  cost  $385,  the  horse  costing  $95 
more  than  the  carriage.     What  did  the  horse  cost? 

20.  A  and  B  are  57  miles  apart.  They  travel  toward  each 
other  until  they  meet,  A  traveling  twice  as  many  miles  as  B. 
How  many  miles  did  A  travel? 

21.  A  has  twice  as  many  acres  of  land  as  B,  and  B  has  three 
times  as  many  acres  as  C.  If  all  of  them  have  2400  acres, 
how  many  acres  have  A  and  B  together? 

Exercise  33  —  Oral  Practice 

Do  this  entire  list  in  28  minutes. 

1.  A  merchant  sold  x  yards  of  silk  for  $45.  What  will 
represent  the  cost  per  yard? 

2.  If  a  man  has  a  half-dollars  and  b  quarters,  how  many 
cents  has  he?     How  many  dollars? 

3.  Indicate  the  sum  of  a  and  b,  diminished  by  c.  The 
sum  of  dx  and  x,  diminished  by  y. 

4.  What  will  represent  the  sum  of  three  consecutive 
numbers  of  which  s  is  the  smallest?  Of  which  s  is  the  middle 
number? 


66  ELEMENTARY  ALGEBRA 

6.  If  there  are  x  tens  and  y  units  in  a  number,  what  will 
represent  the  number? 

6.  How  much  butter,  at  h  cents  a  pound,  will  pay  for  n 
pounds  of  tea  at  60  cents  a  pound? 

7.  What  will  denote  the  number  of  square  feet  in  a  piece 
of  paper  I  yards  long  and  w  feet  wide? 

4 

8.  A  farmer  received  x  dollars  for  sheep  which  he  sold  at 
y  dollars  a  head.     How  many  did  he  sell? 

9.  Find  the  value  of  a  bushels  of  apples  at  m  cents  a  peck 
and  h  bushels  of  pears  at  n  cents  a  peck. 

10.  If  a  represents  an  integer,  when  does  a+1  represent  an 
even  number?     When  an  odd  number? 

11.  If  the  difference  between  two  numbers  is  45  and  the 
larger  one  is  x,  what  is  the  smaller  number? 

12.  What  will  represent  the  sum  of  three  consecutive 
even  numbers  of  which  s  is  the  smallest?     s  the  largest? 

13.  The  sum  of  two  numbers  is  175,  and  the  difference 
between  them  is  5  times  the  smaller.     Find  the  numbers. 

14.  The  sum  of  the  ages  of  3  boys  is  6x  years.  If  they  live, 
what  will  be  the  sum  of  their  ages  in  8  years? 

CLEARING  EQUATIONS  OF  FRACTIONS 

96.  Clearing  of  Fractions.  An  equation  containing  frac- 
tions must  be  changed  so  as  to  remove  the  fractions  before 
it  can  be  solved.     Observe  that 

fX20=16 

Multiplying  this  fraction  by  20,  a  multiple  of  its  denominator,  the 
product  is  a  whole  number.  Multiplying  any  fraction  by  a  multiple 
of  its  denominator  gives  a  whole  number,  for  the  denominator  cancels 
with  one  factor  of  the  multiplier. 


EQUATIONS  67 

97.  Principle. —  //  any  fraction  is  multiplied  by  a  multiple 
of  its  denominator,  the  product  is  a  whole  number. 

98.  Problem. —  To  clear  of  fractions,  the  equation 

1-10+1+3  =  1-5+1  (1) 

Multiply  both  members  of  this  equation  by  12,  the  least  common 
multiple  of  the  denominators,  by  multiplying  each  term  in  it,  applying 
cancellation  to  the  fractional  terms,  and  the  result  is 

6x-120+3a:+36  =  4a;-60+2x     (Mult.  Axiom)       (2) 

Every  term  in  this  equation  is  a  whole  number.  This 
work  is  called  clearing  an  equation  of  fractions. 

In  describing  this  transformation  of  an  equation,  students  should 
tell  by  what  they  multiply  both  members  of  the  equation,  rather  than 
use  the  expression,  clearing  of  fractions,  i.e.,  they  should  say:  **by  the 
use  of  the  multiplication  axiom,"  etc. 

Solving  equation  (2),  x  =  S 

Checking  in  (l):     f-lO+f +3  =  |-5+f 
4-10+2+3  =  %^— 5 
or,     -1  =  -1 


Exercise  34 
Clear  of  fractions,  solve,  and  check  the  following: 

.|-,+|  =  3+|  q--+S  =  S-J 

3.  1+2+1=4+1  .  |+.i  =  38+| 


68  ELEMENTARY   ALGEBRA 

Exercise  35  —  Problems  and  Equations 
Solve  and  check  the  following: 

1.  A  woman  bought  silk  at  $2  a  yard  and  had  $14  left. 
Twice  as  many  yards  at  $1.50  a  yard  would  have  cost  $4 
more  than  she  had.     Find  the  cost  of  the  silk  bought. 

Let  n  =  the  number  of  yards  she  bought; 
then  2n  +  14—  the  number  of  dollars  she  had, 
and  3n  —  4  =  the  number  of  dollars  she  had. 
3w-4  =  2n  +  14 

n  =  18,  and  2n  =  36  ($36,  cost  of  silk.) 
Check:  3- 18-4  =  2- 18-|-14 

50  =  50 

2.  A  has  twice  as  many  sheep  as  B  and  35  less  than  C.  If 
all  have  635,  how  many  has  A? 

^    X  .  ^     X     „     X  2s       is        ,  3s+5 

3.  4  +  6-3  =  7--  4.  _+2^+-  =  s+^3- 

6.  A  boy  has  J  as  many  5-cent  pieces  as  dimes.  If  he  has 
$9  in  all,  how  many  coins  has  he? 

8.  A  and  B  together  earn  $200  a  month;  A  and  C,  $215; 
B  and  C,  $235.     How  much  do  all  earn? 

11.  Two  horses  cost  $350,  one  costing  1 J  times  as  much  as 
the  other.     Find  the  cost  of  each. 

12.  Half  of  a  number,  diminished  by  6,  is  equal  to  ^ 
of  the  number,  increased  by  2.     Find  the  number. 

13.  The  sum  of  the  ages  of  mother  and  daughter  is  48 
years,  and  the  difference  between  their  ages  is  four  times  the 
daughter's  age.     Find  the  mother's  age. 


EQUATIONS  69 

14.  A  man  bought  48  sheep  and  had  $22  left.  If  he  had 
bought  56  at  the  same  price,  he  would  have  needed  $14 
more  to  pay  for  them.     How  much  money  did  he  have? 

16.  A  horse  and  harness  cost  $260,  the  horse  costing  5^ 
times  as  much  as  the  harness.     Find  the  cost  of  each. 

16.  8x-12-(3a:-15)  =  5x+33-(15+3a;) 

17.  A,  B,  and  C  together  earn  $3700.  A  earns  $300  more 
than  B  and  $400  less  than  C.     How  much  does  C  earn? 

18.  5n-(7n-fl6)  +  10  =  12-(9n-30)-h3n 

19.  A  has  J  as  much  money  as  B,  and  C  has  2f  times  as 
much  as  A  and  B  together.  If  C's  money  exceeds  B's  by 
$2800,  how  much  have  all? 

20.  6s+14-(5s+20)=2s+13-(6s-16) 

21.  A  father  and  son  earn  $126  a  month.  If  the  son's 
wages  were  doubled,  he  would  receive  only  $18  less  than  his 
father.     How  much  does  the  son  receive? 

22.  22/ -(32/+ 18) -12  =10 -(42/ -24) -5!/ 

23.  Three  men  raised  2040  bushels  of  corn.  A  raised  three 
time»  as  many  bushels  as  B  and  165  bushels  more  than  C. 
How  many  bushels  did  A  and  B  together  raise? 

24..  Ix-  (14+6a:- 18)  =55-  (5a;-48+3x) 

25.  A  has  \  as  many  sheep  as  B.  If  A  should  double  his 
flock  and  B  should  sell  120  to  C,  A  and  B  would  then  have 
the  same  number.     How  many  sheep  has  B? 

26.  Frank  has  \  as  many  marbles  as  John.  If  John  loses 
186  and  Frank  loses  12,  they  will  each  have  the  same  number 
left.     How  many  marbles  have  both? 

27.  In  a  company  of  112  persons,  it  was  found  that  there 
were  twice  as  many  women  as  men  and  twice  as  many 
children  as  women.     How  many  children  were  there? 


70  ELEMENTARY  ALGEBRA 

GENERAL  REVIEW 
Exercise  36  —  Oral  Review 
Do  this  page  in  15  minutes. 

1.  Express  six  times  the  product  of  a  and  h,  increased  by  3 
times  the  sum  of  x  and  y. 

2.  What  will  represent  the  sum  of  4  consecutive  numbers 
of  which  X  is  the  largest? 

3.  A  man's  capital  doubled  for  3  successive  years,  when  it 
was  $16,800.     How  much  had  he  at  first? 

4.  What  is  the  age  of  a  man  who  y  years  ago  was  a  times 
the  age  of  a  boy  whose  age  was  x  years? 

6.  How  many  square  yards  are  there  in  the  walls  of  a 
room  3x  feet  by  2x  feet  and  y  feet  high? 

6.  What  will  represent  the  sum  of  5  consecutive  numbers 
of  which  m  is  the  middle  one? 

7.  A  boy  had  a  dollars.     He  earned  h  dollars  and  then 
spent  c  dollars.     How  much  did  he  have  left? 

8.  If  one  number  is  n  and  another  number  is  4  times  as 
large,  what  is  the  sum  of  the  numbers? 

9.  A  farm  cost  3  times  as  much  as  a  house.     If  the*farm 
cost  $6200  more  than  the  house,  what  did  both  cost? 

10.  If  a  field  is  x  rods  square,  how  many  rods  of  fence  will 
be  required  to  enclose  it  and  divide  it  into  4  squares?     ^ 

11.  A  girl  has  x  quarters,  y  dimes,  and  z  nickels.     Give  an 
expression  to  denote  how  many  dollars  she  has. 

12.  What  will  denote  the  number  of  feet  in  the  perimeter 
of  a  rectangle  6x  feet  long  and  3a:  feet  wide? 

13.  A  man  bought  x  sheep  at  a  dollars  a  head  and  had  b 
dollars  left.     How  much  money  had  he  at  first? 

14.  A  house  cost  3  times  as  much  as  the  lot,  one  costing 
$5000  less  than  the  other.     What  did  both  cost? 


GENERAL  REVIEW  71 

Exercise  37  —  Written  Review 
Solve  all  the  problems  of  this  page  in  20  minutes. 

1.  A's  age  is  to  B's  as  5  to  7,  and  the  sum  of  their  ages 
is  132  years.     Find  the  age  of  each. 

Let  5n  =  the  number  of  years  in  A's  age, 
and       7n  =  the  number  of  years  in  B's  age. 
5n+7n  =  132 
The  pupil  will  understand  that  the  number  sought  is  not  the  value 
of  n,  but  the  numbers  represented  by  5w  and  7n. 

2.  B's  age  is  to  A's  as  4  to  7,  and  the  difference  between 
their  ages  is  27  years.     Find  A's  age. 

3.  Seven  boys  and  12  men  earn  $275  a  week.  If  each  man 
earns  4  times  as  much  as  each  boy,  how  much  do  the  7 
boys  earn  per  week? 

4.  A  has  3  times  as  many  cows  as  B;  but  if  A  should  sell 
6  to  B,  they  would  then  have  the  same  number.  How  many 
cows  have  both  men? 

5.  Three  men  engage  in  business  with  a  capital  of  $11,000. 
B  invests  half  as  much  as  A  and  $200  more  than  C.  How 
much  have  A  and  B  invested? 

6.  A,  B,  C,  and  D  have  290  sheep.  B  has  15  more  than 
A,  C  has  15  more  than  B,  and  D  has  15  more  than  C.  How 
many  have  A  and  B? 

7.  Three  men  raised  1684  bushels  of  oats.  A  raised  3  times 
as  many  bushels  as  C,  and  185  bushels  more  than  B.  How 
many  bushels  did  B  and  C  raise? 

8.  A  horse,  carriage,  and  harness  cost  $350.  The  horse 
cost  $95  more  than  the  harness,  and  the  carriage  $35  less  than 
the  horse.     Find  the  cost  of  the  horse. 

9.  A  boy  bought  oranges  at  3^  apiece  and  had  20^  left. 
At  bi  apiece,  he  would  have  needed  16j^  more  to  pay  for 
them.     How  many  did  he  buy? 


72  ELEMENTARY   ALGEBRA 

Exercise  38  —  Questions  and  Problems 

1.  Define    algebraic    expression;    term;    monomial;    poly- 
nomial; similar  terms;  value  of  an  algebraic  expression. 

2.  From  what  expression  must  9x-\-Qy  —  5z  be  subtracted 
to  give —4x  —  3y-j-5z? 

3.  How  are  8-9  and  6-9  added  in  arithmetic?      Why 
cannot  9a  and  8a  be  added  in  the  same  manner? 

4.  What    are    the    factors    of    a    number?     Distinguish 
between  the  parts  of  a  number  and  the  factors  of  it. 

6.  How  may  numbers  which  are  expressed  by  2  factors 
be  added,  if  they  have  a  common  factor? 

,    6.  How  is  4-8  subtracted  from  9-8  in  arithmetic?     In 
what  other  way  might  it  be  subtracted? 

7.  What  expression  must  be  subtracted  from  7a  — 56+ 4c 
to  give  9a +66 -2c? 

8.  Define  identity;  equation  of  condition.     Give  examples 
and  show  how  they  differ. 

9.  SimpUfy  9a-  (2b-c)-\-2d-  (5a+36)+4c-2c^,  and  find 
its  value  if  a  =  8,  b=  —4,  c=  —  5. 

10.  Add    a(a+6)+2(6+c)+2(6-c),     -3(6+c)+2(a+6) 
+6(6  — c),  and  a(6+c)— a(a+6)— 4(6  — c). 

11.  How  is  the  correctness  of  subtraction  proved  in  arith- 
metic?    Is  the  same  test  applicable  in  algebra? 

12.  Subtract  2z-]-x  —  2u-\-y-\-7  from  the  sum  of  4:X  —  2y-\- 
5z—u  and  Sy+Q—4z  —  2x. 

13.  Perform  two  different  operations  on  an  equation  so  that 
one  term  shall  be  transposed  from  each  member. 

14.  Describe  four  operations  which  change  the  form  and 
value  of  the  members  of  an  equation,  but  not  their  equality. 


GENERAL  REVIEW  73 

15.  What  expression  must  be  added  to  6a;  — 5^+ 4^;  to  give 

9x-\-4y-7z? 

16.  In  the  identity,  5x-{-Sx  —  2x=10x—4:X,  what  number 
does  X  represent? 

17.  From  36  — 2cH-5d— 4e  subtract  the  sum  of  3d  —  5e—4c 

+26  and  c+e  +  2d-4h, 

18.  How  do  you  subtract  one  term  from  another,  if  the 
two  terms  are  partly  similar? 

19.  To  what  expression  must  8a— 46+ 9c  be  added  to  give 
5a+26-6c?     To  give  0? 

20.  Define  root  of  an  equation.     How  do  you  determine 
whether  a  number  is  a  root  of  an  equation? 

21.  Subtract  2a— 4b +5  from  0,  and  add  the  difference  to 
the  sum  of  5a  — 3c  and  unity. 

22.  Name  the  different  steps  in  the  solution  of  a  problem  by 
the  use  of  an  equation.     Illustrate. 

23.  What  must  be  true  of  two  number  expressions  in  order 
that  we  may  place  them  equal  to  form  an  equation? 

24.  State  the  principle  for  enclosing  two  or  more  terms  of  a 
polynomial  in  a  parenthesis.     Illustrate. 

25.  How  are  terms  that  are  partly  similar  added?     Write 
3  terms  that  are  partly  similar  and  add  them. 

26.  From  the  sum  of  2xy-\-3xz  —  yz  and  Sz  —  2xz-^xy  sub- 
tract the  sum  of  xz  —  yz  and  5z-\-Sxy  —  xz. 

27.  Add    a(a-x)-2(a+a;)+a(a-2),    3(a+a;)  +  (a+3)- 

(a  — 2)— a(a  — a;),  and  a(a+3)  — (a+x). 

28.  How  do  you  prove  whether  the  numbers  found  in 
solving  a  problem  satisfy  the  conditions  of  the  problem? 


CHAPTER  VII 

GRAPHING   DATA.      SOLVING   SIMULTANEOUS 
EQUATIONS  GRAPHICALLY 


GRAPHING  DATA 

99.  Graphing,  as  was  illustrated  in  Chapter  V,  means 
representing  by  pictures  and  diagrams. 

100.  The  diagrams  and  exercises  below  show  how  to 
picture  laws  that  connect  two  sets  of  related  numbers,  such 
as  prices  and  dates,  temperatures  and  times,  etc.,  when  the 
laws  cannot  be  expressed  as  equations,  as  well  as  when  they 
can  be  so  expressed. 

Exercise  39 

1.  In  a  newspaper  of  January,  1916,  the 
prices  of  wheat  from  Jan.  10  to  15  on  a 
Board  of  Trade  were  given  as  in  the  figure. 
The  numbers  along  the  horizontal  are  the 
dates,    and    those    along    the    vertical,    the 

prices  per  bushel.     What  was  the  price  of  wheat  on  Jan. 

10?     On  Jan.  11?     12?     13?     14?     15? 

2.  On  what  date  was  the  price  highest?  Lowest?  Be- 
tween what  dates  did  the  price  change  most? 

3.  The  average  price  per  share,  for 
dates  Jan.  8-15,  1916,  of  20  leading 
stocks  of  the  New  York  Stock  Exchange, 
was  as  shown  in  the  figure.  How  much 
did  the  price  fall  from  Jan.  8  to  Jan.  10? 
Between  what  other  dates  did  the  price 
fall?  Rise?  What  day  was  the  rise 
greatest?     The  fall  greatest? 


II    13    13  la 
JANUARY 


!  $93  55| r 

93  00  r— 

1  75  -A 


II    12    13   in 

JANUARV 


74 


GRAPHING   DATA 


75 


4.  What  was  the  average  price  of  these  stocks  on  Jan. 
11?     On  Jan.  14?     On  Jan.  15? 


JAN.1§ 


+  7° 
+  6° 
+  5° 
+  4° 

-4-3  = 


_2  = 
-  3° 


6789 lO 11121     23    45     6 


6.  The  hourly  temperatures 
from  6  a.  m.  to  6  p.  m.  of  Jan. 
18,  1916,  in  Chicago,  were  as 
shown  in  the  figure.  Observe 
the  degree-numbers  along  the 
vertical  and  the  hour-numbers 
along  the  horizontal,  and  give 
the  temperature  at  6  a.  m.;  at 
9  a.  m.;  at  12  m.;  at  2  p.  m.; 
at  6  p.  m. 

6.    At    what    hour  was    the 

te:nperature  lowest  on  Jan.  18? 

At  what  hours  highest?     When  does  the    graph  show  the 
temperature  stationary? 

In  graphing  temperatures,  the  lines  connecting  the  points  that  repre- 
sent hourly  readings  do  not  represent  the  temperatures  for  the 
intermediate  points.  The  temperature  was  probably  not  stationary 
at  any  time.  But  from  the  hourly  readings  it  was  apparently 
stationary.  Nevertheless,  the  graphs  give  a  good  notion  of  the  general 
trend  of  the  temperature  for  the  day. 

7.  The  hourly  thermometer  readings  from  6  a.  m.  to  6 
p.  m.  on  Jan.  17,  1916,  in  Chicago,  were: 

A.  M.  M.  p.  M. 


Hours  6,    7,    8,     9,      10,     11,     12,  1,      2,      3,      4,      5,      6 

Reading     +2°,  2°,  0°,-l°,-l°,-2°,-2°,-2°,-2°,-2°,-2°,-2°,-r 

Show  that  the  temperature  line  is 

as  given  in  the  figure.     When  was  +-2 

it  coldest?  Warmest?   When  grow-  o 

ing   colder?      Warmer?     When  ^^ 

stationary  by  the  graph?  -a 

See  note  after  problem  6. 


, 

n 

s 

\ 

A 

M 

Ppl     1 

'< 

4-i§s^ 

01 

1   1 

2 

2  : 

}    4    5    6 

^     ' 

\ 

>-C 

)-C 

)-6-< 

:tf' 

^ 

■■ 

1 

76 


ELEMENTARY  ALGEBRA 


+  2 

JA^ 

(.1 

7 

JAN 

ie 

■ 

P    M 

A_^M 

°< 

L|.^°J 

V 

2 

2 

: 

^ 

s 

>    € 

)-< 

/ 

] 

r  ) 

k 

, 

.;^ 

f 

'    ~ 

,, 

^ 

y 

)-C 

N 

>- 

^ 

s 

" 

1 

1 

2's 

<    7 

g    6 
W     5 

-      1 


8.  The  hourly  temperature  curve 
from  6  p.  m.  Jan.  17  to  6  a.  m. 
Jan.  18,  1916,  was  as  shown  in 
the  figure.  What  was  the  ther- 
mometer reading  at  7  p.  m.?  At 
8,  9,  10,  and  11  p.  m.?  At  mid- 
night? At  3  a.  m.  of  the  18th? 
At  5  a.  m.?     At  6  a.  m.? 

9.  From  Jan.  17,  6  p.  m.  to  Jan.  18,  6  a.  m.  when  was 
it  growing  warmer?    Colder?   When  stationary  by  the  graph? 

10.  A  class  studied  the  move- 
ment  of   a   snail   by   having   ft 
crawl    along   a   foot-rule.      The 
observing  time,  in  minutes,  was 
written    along    the    horizontal, 
and    the    distances    crawled,    in 
inches,   along  a  vertical,   giving 
a  picture  of  the  snail's  rate  of 
crawling  as  in  the  figure.     How 
far  had  the  snail  crawled  the  first  minute?     The  first  2 
min.?     In    4    min.?     In    6    min.?     In     12    min.?     What 
minute  did  it  crawl  most  rapidly?     Most  slowly? 
11.  The  daily  growths  of  a  tulip  in  inches  were: 

Day 0       123      4.     56789       10 

Height 0       Ij     3      3|    6      8      8^    9     lOj  llj'   12 


Mark  off  the  days  along  a 
horizontal  and  the  growths  along 
verticals  through  1,  2,  3,  etc., 
using  a  scale  of  1  short  side  to 
1  inch,  and  draw  a  broken  line 
connecting  the  points. 

What  was  the  least  growth  on 
any  day?    The  greatest  growth? 


,Jr^ 

i^ 

ii  t 

■y<A 

A  1 

'                     y 

lA 

-i^  - 

M 

,2:- 

234    5    678    9  ion 
MINUTES 


3    4    S   6    7    8   9 
DAYS 


GRAPHING   DATA 


77 


12.  Using  a   convenient   scale   and   calling   the  verticals 
age-lines,  graph  these  average  heights  of  boys: 

Ages ....     2       46       8     10     12      14      16      18  20  years 

Heights..  1.6   2.6   3    3.5   4.0    4.8     5.2     5.5     5.6  5.7    feet 


_ 

_ 

_ 



' 

— 

r 

1 — ' 

_ 

-^ 

q 

[I 

_ 







in  _ 

— 

- 

— 

— 

— 

— 

^ 

F 

^ 

" 

r 

" 

^^-^ 

— 

— 

o 

^ 

.-s 

> 

lU 

^ 

■-■ 

^ 

U 

_ 

L_ 

to       12        14        16        18       20 


13.  At  what  age  does  the  average  boy  grow  most  rapidly? 

14.  Graph  these  average  heights  of  girls : 

Ages 2      4    6       8     10     12      14      16      18     20  years 

Heights..  1.6   2.6   3    3.5   3.7    4.5     4.8     5.1     5.3    5.4     feet 
See  the  figure  above. 

15.  At  what  age  does  the  average  girl  grow  most  rapidly? 

16.  A  varying  rectangle  5  units  wide,  has  the  following 
lengths  and  areas: 


Lengths   0 

Areas 0 


2 

10 


3 
15 


4 
20 


5  6,  etc. 

25        30,  etc. 


Graph  these  lengths  along  a  horizontal 
and  areas  along  verticals,  and  connect  the 
points. 

In  this  case  we  can  show  the  law  of  the 
areas  by  the  picture,  and  we  can  also  express 
this  law  algebraically  as  an  equation,  thus, 

A=5L 


''A 

/ 

■^ 

/ 

/ 

b^ 

A 

G 

u 

_ 

— 1 

O     1     2    3     4    5    6 
LENGTHS 
"Vertical  Scale 
1   space  =  6 


Pupils   may   find  tables   of   values  in  books,  daily  papers,   trade 
journals,  etc.,  and  graph  them. 


78 


ELEMENTARY  ALGEBRA 


17.  The  rate  per  cent  being  5,  graph  the  following  percent- 
ages and  bases : 

Base $100,  $200,  $300,  $400,  $500,  $600,  $700,  $800,  $900,  $1000 

Percentage..     $5,    $10,    $15,    $20,    $25,    $30,    $35,    $40,    $45,      $50 


S50 
45 

M 

■  ^ 

^ 

-f^ 

y^ 

O 
<35 

-30 

-25 

,^ 

^ 

.^^ 

>\ 

y^ 

^ 

■^ 

^ 

.^ 

> 

)X- 

->1 

^ 

>- 

) 

_ 

SIOO  200   300  400  500  600  700  800   900  1  OOO 

Base   Scale 

1   horizontal   space  =  $50 

1   vertical    space  =  $5 

There  is  also  an  algebraic  expression  of  this  law,  thus, 


P  =  5.—  orp: 


h_ 
20 


18.  If  now  we  take  an  algebraic  law  like  y  =  x-\-\,  we  may 
substitute  successive  values  of  x,  right  and  left  from  0,  and 
calculate  the  corresponding  values  of  y,  thus : 

x=l,     2,     3,     4,     5,     0,     -1,     -2,     -3,     -4,     -5,  etc. 
2/  =  2,     3,     4,     5,     6,     1,        0,     -1,     -2,     -3,     -4,  etc. 

Mark  off  the  x-values  along  the 
horizontal,  to  the  right  if  positive, 
and  to  the  left  if  negative.  Measure, 
to  a  convenient  scale,  the  corre- 
sponding ^/-values  on  the  verticals, 
upward  if  positive  and  downward 
if  negative.  Connect  the  points 
with  a  line.  This  line  is  the  graph 
Graphof  y=a;+l  oiy  =  x-\-l. 


\ 

1/ 

i 

/ 

e 

r 

•>> 

^ 

/ 

o 

■X- 

aa 

-: 

'/ 

-> 

1/ 

/ 

- 

/^ 

/ 

) 

GRAPHING   DATA 


79 


19.  Graph  the  algebraic  law  y^x^,  by  substituting 
successive  values  for  x  and  calculating  in  y  =  x'^,  the 
corresponding  values  of  y. 

x  =  0,    1,    2,    3,      4,      5,     -1,     -2,     -3,     -4,     -5,  etc. 


y  =  0,    I,    4,    9,    16,    25, 


9,      16,      25,  etc. 


k 

1 

1 

i 

K 

rjo 

A 

> 

DO 

.1 

\ 

80 

/ 

\ 

J 

1° 
SO 

\ 

( 

\ 

Jn 

/ 

\ 

;'o 

^ 

^ 

^f 

Jo 

} 

Y 

- 

y\ 

L 

'o 

*^ 

Y 

,._ 

- 

Y. 

lo 

1 

-y 

if 

1  ' 

i> 

rd° 

r 

^^ 

.^-^ 

1 

-1 

0-9 -t 

3-7-€ 

-1 

-4- 

•?-? 

\  ?H  = 

6    7     8    9  1 

o 

Graph  of  y=x^ 

Scale 
1  horizontal  space  =  1 
1  vertical  space  =  10 

Mark  the  x-  and  i/-values  off  on  horizontal  and  verticals. 

20.  Graph  the  algebraic  \awx^-\-y^  =  25,  or  ?/  =  ±  \/25  —  x^* 
by  calculating  values  of  y  and  plotting  points  as  above : 

5,  etc. 
0,  etc. 


x=     0 


1 

:4.9  + 


2 

4.6+ 


4     5-1 
3     0    ±4.9 


2 
4.6 


3  -4 

4  ±3 


^ 

— 

r^ 

i^ 

1^ 

f^ 

s^ 

— 

/ 

'^ 

N 

-2 

\ 

/ 

>^ 

If 

-fe- 

1- 

3  - 

2- 

3    i 

I    1 

V 

/^ 

v 

^ 

J 

\ 

J 

' 

^V 

y 

k^ 

u 

^ 

Graph  of  ?/  =  ±  V 25-2:2 
or  x2+z/2  =  25 

*The  expression  V  25— a;^  means  the  square  root  of  25— x^.  The 
sign  ±  means  that  the  number  calculated  for  V  25— x-  may  be  either 
positive  or  negative. 


80  ELEMENTARY  ALGEBRA 

101.  From  the  above  problems  it  is  seen  that  a  group  of 
facts  expressed  by  two  different  sets  of  connected  numbers, 
like  dates  and  prices,  times  and  temperatures,  ages  and 
heights,  a;-values  and  ^/-values  in  an  equation,  may  be 
pictured,  or  graphed.  This  is  generally  done  by  measuring 
off  the  numbers  of  one  set  horizontally  and  of  the  other  set 
vertically,  locating  points,  and  then  connecting  the  points. 

102.  Problems  18,  19,  and  20  have  shown  the  following 
important  facts: 

1.  A  single  equation  in  two  unknowns  is  satisfied  by  many 
pairs  of  values  of  the  unknowns. 

2.  By  measuring  off  x-values  horizontally  and  2/- values 
vertically  to  suitable  scales,  locating  points  and  connecting 
them,  equations  may  give  either  straight  or  curved  line 
graphs,  or  pictures. 

3.  Every  pair  of  values  of  x  and  y  that  satisfies  a  given 
equation  gives  a  point-picture  that  lies  on  one  and  the  same 
line  or  curve. 

4.  It  is  easy  to  see  that  the  x-  and  ^/-distance  of  any 
point  on  the  curve  from  the  chosen  reference  lines,  would, 
if  substituted,  satisfy  the  equation  that  gave  the  graph. 

103.  In  problem  18  the  graph  of  y  =  x-\-l  was  found  to  be 
a  straight  line.  This  could  be  shown  by  stretching  a  string 
along  the  row  of  points.  Any  equation  in  two  unknowns  in 
which  each  unknown  has  the  exponent*  1  (as  3x  — 2t/=  1)  gives  a 
straight-line  graph.  Knowing  this,  it  is  easy  to  draw  graphs 
of  such  equations  by  merely  ch'oosing  two  values  of  x, 
calculating  the  corresponding  two  values  for  y,  locating  the 
two  points,  and  drawing  a  straight  line  through  the  two 
points  with  a  ruler. 

*With  numbers  like  x,  x^,  y,  y^,  the  small  number  written  (or 
understood)  at  the  right  and  above  the  letter  is  called  an  exponent. 
When  no  number  is  written,  as  with  x,  or  n,  or  y,  1  is  understood  to 
be  the  exponent,  just  as  though  the  written  forms  were,  x^,  or  n^,  or  y^. 


GRAPHING   DATA 


81 


,Ji  1 

^ 

^  1 

"y 

0 

f 

f 

5- 

4- 

3-?- 

J  f 

/ 

A 

/ 

-- 

_ 

L"^ 

_ 

_ 

Y' 


A  third  point  may  well  be  calculated  and  located  as  a 
check  on  the  work. 

It  is  best  not  to  take  the  values  of  x  too  near  together,  as 
it  is  difficult  to  draw  a  line  accurately  through  two  very  near 
points. 

104.  Linear  Equations.  Since  equations  in  two  unknowns 
both  with  exponent  1,  have  straight-line  graphs,  they  are 
commonly  called  linear  equations. 

Y 

1.  Graph  the  linear  equation 

?>x-2y  =  l. 
Take      x  =  {),         +3,     -2,  and 
compute    y=-\,     +4,     -3|. 

The  number-pairs  for  the  points  are 
written  thus : 

-   (0,  -i),  (3,  4),  (-2,  -3i), 
the   first    number    in    the    parenthesis 
being  the  x- value. 

Graph  the  first  two  points  (0,  —  ^)  and  (3,  4),  as  at  A 
and  5,  draw  a  hne  through  them  with  a  ruler,  and  test 
whether  the  point  (  —  2,  —3^)  lies  on  the  line,  as  at  C 

2.  In  a  similar  way  graph  each  of  the  following  equations: 

i.  y  =  x-2  2.  y  =  x-^  3.  y  =  2x 

4.  y  =  4:-x  5.  y  =  2x-\  6.  y  =  2x-\-?> 

7.  x^-2y  =  ^  8.  2x-y  =  4.  9.  3a;-4t/  =  4 

105.  We  have  just  seen  that  one  linear  equation  in  two 
unknowns  is  satisfied  by  many  pairs  of  values  of  x  and  y. 
But  two  linear  equations  in  two  unknowns,  such  as 

2x--^y  =  7 
2y  —  x  =  4: 

can  both  be  satisfied  at  the  same  time  by  only  one  pair  of 
values  of  x  and  y. 


Linear  Equation 
Straight-Line  Graph 
Graph  of  Sx-2y  =  l 


82 


ELEMENTARY  ALGEBRA 


A" 


T^ 

^K 

\ 

S- 

% 

D\-        L^^ 

X<^B 

l.r-'^S- 

*^Ji^    V 

^WG     O          El\ 

^              4--- 

T         ^ 

it         _SJ 

For  example,  graph  2x-\-y  =  7, 

using  a:  =+1,  +3,  and  —1, 
giving  2/=  +5,  +1,  and  +9, 

and  graph  2?/  — x  =  4, 

using  x=     0,  +4,  and  —3, 
giving  y=-\-2,  +4,  and  H-|.     (See  figure.) 

Now,  we  ask,  can  a  point  he  so  as 
to  give  X-  and  ^/-distances  that  will 
satisfy  both  equations? 

The  answer  is  yes.  The  point  of 
intersection,  P,  of  the  graphs  sat- 
isfies the  requirement.  For  the 
point,  P,  x=-\-2  and  y=-\-S,  and 
these  values  satisfy  both  equations. 
Hence,  the  x-  and  ^/-distances  of  the 
point  of  intersection  of  the  graphs 
are  the  graphical  solution  of  the  two 
given  linear  equations.     Since  the 

graphs  cross  at  only  one  point  there  is  only  one  solution 

of  the  pair  of  equations. 

106.  Hence,  two  linear  equations  in  two  unknowns  can  be 
satisfied  by  only  one  pair  of  values  of  the  unknowns. 

SOLVING  SIMULTANEOUS  EQUATIONS  GRAPHICALLY 

107.  Simultaneous  Equations.  Equations  that  can  be 
satisfied  by  the  same  values  of  the  unknowns  are  called 
simultaneous  equations. 

108.  It  is  now  worth  while  to  see  that  not  all  pairs  of 
linear  equations  in  two  unknowns  can  be  satisfied  by  even 
one  pair  of  values  of  the  unknowns. 

Two  or  more  equations  considered  together  are  said  to 
form  a  system. 


Y' 

Simultaneous  Equations 
Intersecting  Graphs 


SOLVING  EQUATIONS   GRAPHICALLY 


83 


1.  Consider  the  system 


1.  2y-x  =  4: 

2.  Qy-Sx  =  Q 


The  graphs  of  the  equations  are  shown  in  the  figure. 
Dividing  2  through  by  3,  gives  2y  —  x  =  2,  and  the  graph  on 
which  this  is  written  is  the  graph  of  6y  —  3x  =  Q.  The  graphs 
are  a  pair  of  parallel  lines.     They  y 

do  not  meet,  and  there  is  no 
point  that  Ues  on  both  graphs. 
This  means  there  is  no  pair  of 
values  of  x  and  y  that  will  satisfy 
both  equations. 


"H 

n 

.M'l 

r^ 

^"^1 

X 

^ 

^ 

^ 

ft  (• 

i>< 

^ 

ni> 

ii> 

>^ 

^^^ 

^ 

^i^ 

r 

^ 

Non-Simultaneous  Equations 
Parallel  Graphs 


109.    Inconsistent    Equations. 

Equations  which  cannot  be  satis- 
fied by  any  pair  of  values  of  the  unknowns  are  called  non- 
simultaneous,  or  inconsistent  equations. 


That  the  equations  of  §  108  are  inconsistent  can  be  seen 
without  graphing,  by  dividing  the  second  through  by  3. 
This  does  not  change  the  relation  between  x  and  y.  Then 
one  equation  says  that  2y—x  =  4:,  and  the  other  that  2y  —  x 
is  at  the  same  time  equal  to  2.  This  is  obviously  absurd. 
The  number,  2y—x,  cannot  at  the  same  time  be  both  4 
and  2. 

110.  For  a  system  of  two  linear  equations  in  two  unknowns 
to  be  capable  of  solution,  the  equations  must  be  simultaneous. 

111.  Dependent  Equations.  It  is,  however,  not  sufficient 
that  the  equations  be  simultaneous.  We  shall  now  see  that 
two  linear  equations  in  two  unknowns  can  fail  to  give  a 
definite  solution  because  they  have  too  many  solutions. 


1.  Consider  the  system. 


1.  2y-x  =  5' 

2.  Qy-3x=15 


84 


ELEMENTARY  ALGEBRA 


1 

V 

^ 

?il>1 

jX 

6^ 

X 

<;, 

°> 

*r 

f^ 

Y^ 

■ 

*L 

^ 

o 

^ 

1 

Both  graphs  are  shown  in  the  figure  as  a  single  line.  They 
coincide.     Every  point  that  is  on  one  is  on  the  other  also. 

Hence,  any  pair  of  values  of  x  and 
y  that  satisfies  one  of  the  equations, 
satisfies  the  other  also.  Dividing 
the  second  equation  through  by  3, 
gives  2z/  — a:  =  5,  which  is  identical 
X  with  equation  1.  One  equation  de- 
pends  on  the  other  in  the  sense  that 
one  can  be  derived  from  the  other 
by  simple  division  by  an  arithmeti- 
cal number. 

Such  equations  are  called  dependent  equations. 

112.  Finally,  for  a  system  of  two  linear  equations  in  two 
unknowns  to  be  capable  of  solution,  the  equations  must  be  both 
simultaneous  and  independent. 


Dependent  Equations 
Coincident  Graphs 


Exercise  40  —  Graphical  Solutions 

Solve  the  following  systems  graphically,  or  in  case  there 
is  no  definite  solution,  tell  whether  the  system  is  inconsistent 
or  dependent: 


4. 


x-y  =  2 
Sx-2y  =  9 

x-\-y  =  5 
x-Sy=l 

2x-5y=15 
5y-2x=-15 


2. 


8. 


(     x+y=l 
\2x-\-5y  =  n 

I  x-{-2y  =  Q 
\2x-\-4y=12 

y  =  2x-3 
x-\-2y  =  14: 


3. 


9. 


f     x+y=2 
\3x+Sy  =  Q 

y  =  x-S 
Sx-6y=ll 


5x-Sy  =  S 
2x+y=10 


The  graphical  way  of  solving  equations  makes  the  mean- 
ing of  solutions  clear;  but  the  algebraic  way  of  the  next 
chapter  is  shorter,  and  as  it  can  be  applied  to  equations  in 
3,  4,  5,  and  even  n  unknowns,  it  is  also  much  more  generally 
useful  than  the  graphical  way. 


CHAPTER  VIII 

SIMULTANEOUS    SIMPLE    EQUATIONS.      ELIMINA- 
TION BY  ADDITION  OR  SUBTRACTION 

SIMULTANEOUS  SIMPLE  EQUATIONS 

113.  A  determinate  equation  is  an  equation  which  has  one 
root,  or  a  limited  number  of  roots,  as, 

114.  An  indeterminate  equation  is  an  equation  which  has 
an  unlimited  number  of  roots. 

Consider  2x-{-2y  =  12 

Any  value  may  be  assigned  to  x  in  this  equation  and  a 
value  of  y  found  that  will  satisfy  the  equation. 

For  example,  when  x  =  l,  y  =  5;  when  x  =  2,  y  =  4:;  when 
X  =  3, 2/  =  3 ;  and  so  on  indefinitely. 

115.  Now  consider       2x  —  2y=  —4: 

Any  value  may  be  assigned  to  x  in  this  equation  and  a 
value  of  y  found  that  will  satisfy  the  equation. 

For  example,  when  x=l,  y  =  S;  when  x  =  2,  2/  =  4;  when 
x  =  S,y  =  5;  and  so  on  indefinitely. 

It  is  evident  that  every  simple  equation  containing  two  or 
more  unknown  numbers  is  indeterminate. 

But  there  is  one  set  of  values,  and  only  one,  that  satisfies 
both  equations,  2x-{-2y  =  12  and  2x  —  2y=  —4,  and  these 
values  are  x  =  2  and  i/  =  4. 

85 


86  ELEMENTARY  ALGEBRA 

116.  Independent  equations  are  equations  which  cannot  be 
derived  one  from  the  other  by  addition  of,  or  multipH- 
cation  or  division  by  a  positive  or  negative  arithmetical 
number. 

The  equations  given  above  are  independent,  for  one  cannot  be 
derived  from  the  other  by  simple  multiplication  and  division.     So  also 

are  4a:+3^  =  28  and  2x+3y  =  U. 

117.  A  system  of  equations  is  two  or  more  equations 
involving  two  or  more  unknown  numbers,  as, 

x+2y  =  d2  (2x-\-Sy  =  SQ 

x-2y  =  12  \Qx-2y  =  20 

By  a  set  of  roots  is  meant  the  values  of  the  unknown 
numbers  in  a  system. 

As  has  been  noted,  each  equation  of  a  system,  when  taken 
by  itself,  is  indeterminate.  It  was  noted,  also,  that  only  one 
set  of  roots  will  satisfy  two  independent  equations.  In  the 
two  systems  above,  a:  =  22  and  y  =  5  in  the  first  and  x  —  Q  and 
y  =  S  in  the  second,  were  the  sets  of  roots. 

Simultaneous  simple  equations  were  solved  graphicallj^ 
in  Chapter  VII.     They  will  now  be  solved  algebraically. 

To  solve  two  simultaneous  equations  containing  two 
unknown  numbers,  it  is  necessary  to  obtain  from  them  a 
single  equation  containing  but  one  unknown  number. 

This  can  be  done  only  in  case  the  equations  are  indepen- 
dent as  well  as  simultaneous;  see  §  112. 

118.  Elimination  is  the  process  of  combining  two  or  more 
simultaneous  equations  containing  two  or  more  unknown 
numbers  in  such  a  way  as  to  obtain  a  single  equation  in  which 
one  of  the  unknown  numbers  does  not  appear. 


ELIMINATION  BY  ADDITION  OR  SUBTRACTION       87 
ELIMINATION   BY   ADDITION   OR   SUBTRACTION 

119.  The  following  examples  indicate  the  method  of  elimi- 
nation by  addition  and  by  subtraction. 

Solve  the  systems: 

x-\-y=  8  (1)  f3a;+32/  =  9  (1) 

•  ^    x-y=  6  (2)  ;  \3x+  y  =  b  (2) 


2x       =14  22/  =  4 

X       =7  y=2 

We  add  (2)  to  (1),  member  to  We    subtract     (2)     from     (1) 

member,  eliminating  y,  and  then  eliminating  x,  and  then  find  the 
find  the  value  of  x.  value  of  y. 

We  then  substitute  these  values  in  one  of  the  equations  of  the  sys- 
tem that  gave  it,  and  find  the  value  of  the  other  unknown  number. 

From  (1).     y=l  From  (2).     x=l 

checking  1^+^  =  7  +  1=8  (1)  p+3^  =  3. 1  +  3- 2  =  9  (1) 

cneckmg  \^_^^^_^^q  (2)         cneckmg  ^^^^  ^  =  3.i_|-  2   =5  (2) 

In  example  3,  given  below,  we  multiply  both  members  of 
(2)  by  2  and  eliminate  y  by  subtracting  (3)  from  (1). 

3.  9a:+4i/  =  43  (1)  4.  Sx-\-2y  =  2l  (1) 

Sx-{-2y  =  17  (2)  2x-\-3y  =  19  (2) 

9x+4y  =  4S  (1)  Qx-\-4y  =  42  (3) 

6a;+4!/  =  34  (3)  6x+9y  =  57  (4) 

Sx         =9  5y  =  15 

In  example  4,  we  multiply  (1)  by  2  and  (2)  by  3  and 
eliminate  x  by  subtracting  (3)  from  (4). 

120.  Rule. —  Determine  first  which  of  the  two  unknown 
numbers  it  is  more  convenient  to  eliminate. 

By  the  multiplication  axiom,  §15,  make  the  coefficients  of 
that  unknown  number  the  same  in  both  equations. 

If  the  signs  of  the  terms  to  be  eliminated  are  unlike,  add 
the  equations,  member  to  member;  if  alike,  subtract  one  equa- 
tion from  the  other,  member  from  member. 


88  ELEMENTARY   ALGEBRA 

Exercise  41 

Solve  the  following  equations,  checking  some  of  them : 


\Sx-2y  =  4: 


Ax  —  5y=l 

2x-2y  =  2 


5. 


9. 


11. 


(Sx-\-6y  =  Q 
\Qx-3y  =  2 

5x-3y  =  23 
7x-4y  =  SS 

(5x-^Sy  =  SS 
\9y-Sx=15 

7x-Sy  =  29 


13. 


9x-4y  =  S5 

(9x-\-Sy=   12 
\4y-Qx=-l 


2. 


8. 


10. 


12. 


14. 


4x-2y=-8 
x-\-Sy=-9 

(5x-\-'6y=-'i 
\2x-{-  y=-l 

f2x+3i/=-4 
\Sx+5y=-5 


3x-Sy=-d 
7x-Qy=-l 

\4:y-\-5x=—7 

(5x-^Sy=-5 
\Qy+9x=-Q 

f4x+6y=-8 
\8!/+5x=-4 


PROBLEMS 

121.  Solving  Problems.  In  algebra  many  problems  in 
which  two  or  more  numbers  are  to  be  found  can  be  solved 
by  the  use  of  a  single  equation  containing  but  one  unknown 
number,  but  in  many  problems  it  is  more  convenient  to 
introduce  as  many  unknown  numbers  as  there  are  numbers  to 
be  found.  Such  solutions  involve  a  system  of  simultaneous 
equations,  and  to  make  a  solution  possible,  there  must  be  as 
many  independent  equations  as  there  are  unknown  numbers 
used. 


ELIMINATION  BY  ADDITION  OR  SUBTRACTION       89 
Exercise  42  —  Problems  in  Two  Unknowns 

1.  The  larger  of  two  numbers  exceeds  4  times  the  smaller 
by  17,  and  twice  the  larger  exceeds  7  times  the  smaller  by  48. 
Find  the  numbers. 

Let  X  =  the  larger  number, 
and  y  —  the  smaller  number. 

x-4i/  =  17 

2.  If  7  pounds  of  tea  and  5  pounds  of  coffee  cost  $6.50  and 
6  pounds  of  tea  and  10  pounds  of  coffee  at  the  same  prices 
cost  $7,  what  are  the  prices  per  pound? 

Let  X  =the  price  of  the  tea  in  cents, 
and  y  =  the  price  of  the  coffee  in  cents. 

7a;+5i/  =  650 
62: +  10^  =  700 

3.  The  sum  of  two  numbers  is  121,  and  their  difference  is 
25.     Find  the  two  numbers. 

4.  A  boy  has  $2.00  in  dimes  and  nickels,  28  coins  in  all. 
How  many  coins  of  each  kind  has  he? 

5.  Eight  sheep  cost  $12  more  than  9  lambs,  and  5  sheep  and 
3  lambs  cost  $42.     Find  the  price  of  each. 

6.  B's  age  exceeds  A's  age  by  8  years,  and  3  times  A's  age 
exceeds  twice  B's  age  by  28  years.     Find  their  ages. 

7.  Find  two  numbers  such  that  if  4  is  subtracted  from  the 
first  and  8  added  to  the  second,  the  results  are  equal;  while 
if  2  is  subtracted  from  the  first  and  6  from  the  second,  the 
first  remainder  is  twice  the  second. 

8.  Nine  apples  and  8  oranges  cost  59^,  and  at  the  same 
prices  7  apples  and  6  oranges  cost  45^.  Find  the  price  of 
each. 


90  ELEMENTARY  ALGEBRA 

9.  A  man  sold  80  sheep  for  $390,  selling  some  of  them  at 
$4  a  head  and  the  rest  at  $6  a  head.  How  many  sheep  did  he 
sell  at  each  price? 

10.  The  sum  of  the  ages  of  A  and  B  is  92  years.  If  B  were 
twice  as  old  as  he  is,  his  age  would  exceed  A's  age  by  16 
years.     Find  the  age  of  each. 

11.  In  an  election  5163  men  voted  for  two  candidates,  and 
the  candidate  elected  had  a  majority  of  567.  How  many 
votes  did  each  candidate  receive? 

12.  The  sum  of  two  numbers  is  255,  and  f  of  the  larger 
is  equal  to  f  of  the  smaller.  By  how  much  does  the  larger 
number  exceed  the  smaller? 

13.  Twelve  men  and  6  boys  earn  $24  a  day,  and  at  the 
same  daily  wages,  7  men  and  8  boys  would  earn  $16.25  a  day. 
How  much  does  each  man  earn  per  day? 

14.  A  miller  mixes  corn  worth  80^  a  bushel  with  oats  worth 
60^,  making  a  mixture  of  100  bushels  worth  72^  a  bushel. 
How  many  bushels  of  each  does  he  use? 

15.  Eight  years  ago  B  was  3  times  as  old  as  A,  but  if  both 
live  8  years,  B  will  be  only  twice  as  old  as  A.  What  was  the 
age  of  each  8  years  ago? 

16.  A  merchant  sold  48  yards  of  silk  for  $89,  selling  part 
of  it  at  $1.75  a  yard  and  the  rest  at  $2  a  yard.  How  many 
yards  of  the  better  silk  did  he  sell? 

17.  A  has  160  sheep  in  two  fields.  If  he  takes  15  from  the 
first  field  to  the  second,  he  has  the  same  number  in  each 
field.     How  many  are  there  in  each  field? 


CHAPTER  IX 
MULTIPLICATION 

122.  Multiplication  is  the  process  of  taking  one  number 
as  an  addend  a  certain  number  of  times. 

3X5  =  5+5+5  =  15 

123.  The  multiplicand  is  the  number  taken  as  an  addend. 

124.  The  multiplier  is  the  number  which  denotes  how  many 
times  the  multiplicand  is  taken. 

125.  The  product  is  the  result  of  multiplication. 

THE   SIGN    OF   THE   PRODUCT 

126.  Taking  +5  twice  as  an  addend,  we  have  +10;  three 
times,  +15;  four  times,  +20;.  five  times,  +25.     Thus, 

3-  (+5)  =  +15,  4-  (+5)  =  +20,  6-  (+8)  =  +48, 

which  are  the  same  as 

(+3)(+5)  =  +15,       (+4)(+5)  =  +20,       (+6)(+8)  =  +48. 

Taking  —5  twice  as  an  addend,  we  have  —10;  three  times, 
—  15;  four  times,   —20;  five  times,   —25.     Thus, 
4-(-5)  =  -20,  7-(-5)  =  -35,        •    9-(-5)  =  -45, 

which  are  the  same  as 
(+4)(-5)  =  -20,       (+7)(-5)  =  -35,       (+9)(-5)  = -45. 

A  negative  multiplier  means  that  the  product  is  of  the 
opposite  quality  from  what  it  would  be  if  the  multiplier 
were  positive.     Therefore, 

(+5)(-4)  =  -20  (-7)(-6)  =  +42  (-8)(-5)  =  +40 
From  the  foregoing  examples, 

(+6)(+5)  =  +30  (+7)(-5)  =  -35 

(-6)(-6)  =  +36  (-8)(+6)=-48 

91 


92  ELEMENTARY  ALGEBRA 

From  these  results  we  may  derive  a  law  of  signs  for 
multiplying  positive  and  negative  numbers. 

127.  Sign  Law  of  Multiplication.  —  Like  signs  of  two 
numbers  give  a  positive  product,  and  unlike  signs  give  a  negative 
product. 

128.  The  product  of  two  or  more  numbers  must  contain  as 
factors  all  the  factors  of  each  of  the  numbers. 

Thus,  2aX36  =  2-3-a-6  =  6a6 

Exercise  43 
Give  the  products  of  the  following  orally : 


1.   Sx 

2y 

2.  -a6 
3c 

3.      4a 

-3n 

4.  -Qx 

yz 

5.  -5a 
-Sx 

6.    5a 
46 

7.  -xy 

2s 

8.  -6a 

-6c 

9.      46 
-3c 

10.  -76 
-3a 

When  a  term  contains  a  twice  as  a  factor,  it  is  not  written 
aa,  but  a^,  and  is  read :  a  square. 

When  a  term  contains  x  3  times  as  a  factor,  it  is  not 
written  xxx,  but  x^,  and  is  read :  x  cube. 

129.  An  exponent  is  a  symbol  of  number  written  at  the 
right  and  a  little  above  another  symbol  of  number  to  show 
how  many  times  the  latter  is  taken  as  a  factor. 

2ab^&  =  2'a'b'b'C'C'C  =  2abbccc 
This  is  its  signification  only  when  the  exponent  is  a  positive  integer. 

It  must  be  remembered  that  when  no  exponent  is  expressed 
the  exponent  1  is  always  understood.  Thus  abx  means 
a'b'x\ 

Observe  that  a^  =  aXaXaXaXay 

while  5a  =  a-|-a-|-a4-a-|-a 


MULTIPLICATION  93 

Students  should  note  carefully  the  difference  in  meaning  of 
exponent  and  of  coefficient. 

130.  The  sign  of  continuation  is  a  series  of  dots  . . . , 
and  is  read,  and  so  on,  or  and  so  on  to. 

THE  EXPONENT  IN  THE  PRODUCT 

131.  By  §  128,  a^Xa^  =  aaa'aa  =  aaaaa  =  a^ 

In  this  particular  example,  the  exponents  of  a  in  multi- 
plicand and  multiplier  are  added.  This  illustrates  a  law 
of  multiplication. 

The  student  should  understand  here  that  to  prove  any 
general  law,  general  numbers  must  be  used. 

To  prove  that  this  law  of  multiplication  is  general  for 
any  positive  integral  exponents,  let  a  represent  any  number 
and  m  and  n  any  positive  integral  exponents.     Then,  by  §  129, 

a"'  =  a'a'a'a'a  . .  .torn  factors; 
and  a""  =  a- a- a* a- a. .  .ton factors. 

The  product,  a*"  times  a"",  must  contain  a  to  m  factors  and 
a  to  n  factors,  or  a  to  (m+n)  factors.     Therefore, 

132.  Law  of  Exponents  for  Multiplication.  —  The  expo- 
nent of  the  product  is  the  sum  of  the  exponents  of  the  factors. 

The  exponents,  m  and  n,  used  in  this  discussion  are  general 
numbers  only  in  the  sense  that  they  denote  any  positive 
integers. 

MULTIPLYING    ONE    MONOMIAL   BY   ANOTHER 

133.  Rule. —  Write  the  sign  of  the  product,  if  negative  {if 
positive  no  sign  need  be  written),  placing  after  it  the  product  of 
the  numerical  factors  and  all  the  different  letters,  giving  each 
letter  an  exponent  which  is  the  sum  of  the  exponents  of  that 
letter  in  the  factors. 


94  ELEMENTARY   ALGEBRA 

Exercise  44 
Give  the  following  products : 

l,Qa^x         2.      SaH         3.      Qxy^  4.-   xy^  5. -7a^b 

-  ax^  7ifz  -2¥c 

8.  -362c  9.      8aa;2  10.  -^xhj 

—  4a2  h  —  \xHj  —  2x^y 

13.      la¥  14       bo^x  15.  -%a?x 

-2a^b  -9xhj  -2hH 


Sx-'y 

-baH 

6.  4a26 
Sab^ 

7.  —iax^ 
4:a^x 

11.8ax2 
5a'b 

12.  -7ax2 
26x2 

There  are  three  important  fundamental  laws  of  multiplica- 
cation  which  it  will  be  well  to  notice  here. 

These  are :  law  of  order,  or  commutative  law;  law  of  group- 
ing, or  associative  law;  and  distributive  law. 

134.  Law  of  Order. —  The  product  of  several  numbers  is  the 
same  in  whatever  order  they  are  used. 

It  is  evident  that 

8-5-3  =  5-3'8  =  3-8-5 
for  each  member  of  this  equality  is  the  same  number. 
In  general  numbers, 

abc  =  b'C-a  =  c-a-b 

135.  Law  of  Grouping. —  The  product  of  several  numbers  is 
the  same  in  whatever  manner  they  are  grouped. 

8- 5- 3  denotes  that  8  is  to  be  multiplied  by  5  and  the 
product  multiplied  by  3 ;  that  is,  8  •  5  •  3  =  (8  •  5)  •  3. 
By  the  law  of  order, 

8-5-3  =  5-3-8  =  3-8-5 
Therefore, 

8-5-3  =  (8-5)-3  =  (5-3)-8=(3-8)-5 
In  general  numbers, 

a*b-c=  (a-b)-c=  (bc)*a=  (a-c)-b 


MULTIPLICATION  95 

136.  Distributive  Law. —  The  product  of  a  'polynomial  and  a 
monomial  is  the  algebraic  sum  of  the  products  obtained  by 
multiplying  each  term  of  the  polynomial  by  the  monomial. 

(8+7)-6  =  8-6+7-6 
In  general  numbers, 

(b+c)a  =  ab+ac 
This  is  called  the  distributive  law,  because  the  multiplier 
is  distributed  over  the  terms  of  the  multiplicand. 

137.  A  power  is  the  product  obtained  by  taking  a  number 
any  number  of  times  as  a  factor. 

138.  A  square,  or  second  power,  is  the  product  obtained  by 
taking  a  number  twice  as  a  factor.     Thus, 

52  =  5.5  =  25  72  =  7-7  =  49  (6a)2  =  6a -60  =  3602 

139.  A  cube,  or  third  power,  is  the  product  obtained  by 
taking  a  number  three  times  as  a  factor. 

53  =  5 . 5 . 5  =  125  (3a2)3  =  3a2  •  3a^  •  Sa^  =  27a« 
The  repeated  factor  is  the  root  of  the  power,  and  the 

exponent  indicating  the  power  is  the  exponent  of  the  power. 
The  product  is  the  power.     Thus, 

exponent 
root  — »  2^  ^  8  '^—  power 

POWERS   OF   MONOMIALS 

140.  To  find  a  power  of  any  number  is  simply  to  find  the 
product  of  two  or  more  equal  factors.     Thus, 

(2a62)4  =  2a¥  -  2ab''  •  2ab''  •  2ab^  =  Ida'^b^ 
By  the  law  of  signs  in  multiplication,  §  127,  all  powers  of 
positive  numbers  and  even  powers  of  negative  numbers  are 
positive;  odd  powers  of  negative  numbers  are  negative. 

141.  Rule. —  (1)  Raise  the  numerical  coefficient  to  the  re- 
quired  power,  (2)  multiply  the  exponent  of  each  letter  by  the 
exponent  of  the  power,  and  (3)  give  the  result  the  proper  sign. 


96 

ELEMENTARY 

'  ALGEBRA 

Exercise  45 

Gh 

^e  these  indicated  powers : 

1. 

{2ay 

2.  (-2c2)3 

3. 

i-^^y 

4. 

(a^x^y 

6. 

{sxy 

6.  (-4a'y 

7. 

i-hr 

8. 

{a^x'y 

9. 

{2yy 

10.  i-zx^y 

11. 

i-la^y 

12. 

(a^x'y 

13. 

{2ay 

14.  (-2a'y 

16. 

i-Wy 

16. 

(xvy 

17. 

{7xy 

18.  i-5x'y 

19. 

i-^x^y 

20. 

{a'b^y 

21. 

{Say 

22.  {-4a^y 

23. 

{-h'y 

24. 

{xvy 

MULTIPLYING   A   POLYNOMIAL   BY   A   MONOMIAL 

142.  Observe  carefully: 

3a'¥-2a^b^+3a''b-2ab^ 
2ab^ 


Qa^¥-4:a^b^-\-Qa^b^-4a^b^ 


143.  Rule. —  Multiply  each  term  of  the  multiplicand  by  the 
multiplier  as  in  multiplication  of  monomials. 


Exercise  46 
Multiply: 

1.  3ax^+4:a^x  by  Sa^x^  2.  Sa'^¥-ab^-\-Sa^¥  by  4a^b^ 

3.  5x'^y-3xy^  by  4a;y  4.  5ahi^-a^n-^a^n^  by  5aV 

5.  3ac3-4a2c  by  5aV  6.  Qa^b^-a¥-\-Sa^b^  by  Qa^¥ 

7.  Qa^x-7ax^  by  3a^x^  8.  5¥c^+b^c-Wc'^  by  Sb^c'^ 


MULTIPLICATION  97 

MULTIPLYING  A  POLYNOMIAL  BY  A  POLYNOMIAL 

144.  It  follows  from  the  distributive  law,  §  136,  that  {a-\-b) 
times  any  number  is  a  times  the  number  plus  b  times  the 
number. 

145.  Rule. —  Multiply  the  multiplicand  by  each  term  of  the 
multiplier  and  add  the  products. 

Observe  carefully: 


Work 

Check 

^x'-2x''y-^Zxy^-y^ 

=  3 

2x^-xy 

=  1 

6x^  —  4:X^y-{-Qx^y^  —  2x'^y^ 

-  Sx^y + 2xY  -  ^^V + xy"^ 
6x^-7x^y-\-SxY-5xY-\-xy^  =  ^ 

The  work  is  checked  by  substituting  x  =  l  and  y  =  l  in  the  multipli- 
cand, multiplier,  and  product. 

It  is  plain  that  since  any  power  of  1  is  1,  i.e.  1^  =  1^  =  1^  =  1^  =  1, 
substituting  1  for  x  and  ij  does  not  check  the  exponents  of  x  and  ij. 

Exercise  47 
Multiply: 

1.  3c2+4c-6-by  2c^-c-\-S 

2.  4a2+5a-3  by  da^-a+5 

3.  Sx^-{-2x^-4x+l  by  2x+4 

4.  a2-f3a6+62  ^y  2a^-Sab+b'' 
6.  x^y  —  x'^y^  —  Sxy^  by  2x'^y—xy^ 

146.  A  polynomial  is  arranged  when  the  exponents  of  some 
letter  increase  or  decrease  with  each  succeeding  term.     Thus, 

120^2— 4+ 2x  is  arranged  when  put  in  the  form  1 20^24- 2x  — 
4,  or  -4+2x+12x2. 

As  a  convenience  for  the  beginner,  the  multiplicand  and  the  multi- 
plier should  be  arranged  with  reference  to  some  letter;  if  possible,  the 
same  letter. 


98  ELEMENTARY   ALGEBRA 

Exercise  48 
Multiply: 

1.  12x2-4+2x  by  4+5x2-3a; 

2.  3a-2+4a2  by  4:a-\-3a^-2a' 

3.  2x-{-3+x^  by  Sx^-2x+S-x^ 

4.  2a6-362+a2  by  2a''+3ab-b'' 

5.  3a2-262-f3c2  by  4a2-262+4c2 

6.  4ac-3a2+2c2  by  2a2-c2+3ac 

7.  a:^+2a;22/H-4x|/2  by  2?/+?/2  — 40^^ 

8.  3a3-3a+2a2-4  by  5-3a2-3a 

9.  2x?/2— x^+Sa;^?/  — ?/^  by  i/  —  dxy-\-x^ 

10.  3a4+2a-3a2-a3+3  by  3a-a2+4 

11.  4a:3-3x-4a;2+l  by  3a:3+x-3a:2-5 

Perform  the  following  indicated  multiplications  and  unite 
results  into  as  few  terms  as  possible:* 

12.  {a+c-{-x)2{a-\-c-x)-{2ac-x^y 

13.  3(a-2c)(a+2c)2-5(a-3c)2+69c2 

14.  2(3a+x)2-(3a-3a;)(3a+3x)-llx2 

15.  (a-2c)(c-3a)-(3a+c)(2c-a)-2ac 

16.  (2x+32/)2(2a;-32/)+2(2a;+32/)2-19a:2/ 

17.  7{x+Sy){x-3y)-{x-5yy-6{x^-4y^) 

18.  4(a:+3)(a;4-2)  +  (a:-6)(a;+4)-3a:(a;+7) 

*First  decide  how  many  terms  there  are  in  each  of  the  given  exercises. 


MULTIPLICATION  99 

Exercise  49  —  Special  Products 

Perform  the  following  indicated  operations  as  rapidly  as 
you  can,  using  pencil  only  when  necessary : 

1.  (a+l)(a-hl)=       2.  (a+5)(a+5)=     3.  ia-\-xy  = 

4.  (2a:  +  l)2=  5.  {7-\-xy=  6.  {x+4){x+S)  = 

7.  (:c+12)(x+3)=     8.  (8+a)(5+a)=      9.  {x-\-a){x+b)  = 


10.  (m+r)(?n+s)  = 

12.  (x^-\-y^)(x^-y^)  = 

14.  (x+y){x'^-xy-\-y^)  = 

16.  (3x+22/)2  = 

18.  {a-h)ia'-2ab-\-¥)  = 

20.  (a:-l)(a:+l)(:c2+l)  = 

22.  (a+6+a:)(a+b-x)  = 

24.  (a-5)(a-5)  = 

26.  (2a:- 1)2  = 

28.  {x+4){x-S)  = 

30.  (8-a)(5-a)  = 

32.  (m+r)(m  — s)  = 

34.  (x^-{-y^y  = 

36.  (x-2/)(a:2+xi/+^2)=. 


11.  (a2+62)(a2-62)  = 

13.  (x4-h2/4)2  = 

15.  (2a:+l)2  = 

17.  {a-b){a''+ah-\-h-)  = 

19.  (x+2/)(a:2+a:2/+2/2)  = 

21.  (a+6)3  = 

23.  (fl-l)(a-l)  = 

25.  (a-a;)2  = 

27.  {7-xy  = 

29.  (a;- 12) (a? -3)  = 

31.  {x-a){x-b)  = 

33.  (a2+62)2  = 

35.  (x^- 2/4)2  = 

37.  (2a:- 1)2  = 


CHAPTER  X 
SIMPLE  EQUATIONS 

147.  The  degree  of  a  term  is  indicated  by  the  sum  of  the 
exponents  of  the  literal  factors. 

Thus,  a^x^  is  a  term  of  the  fourth  degree. 

The  degree  of  a  term  in  any  particular  letter  is  indicated 
by  the  exponent  of  that  letter  in  the  term. 

Thus,  a^x^  is  of  the  second  degree  in  x. 

148.  The  degree  of  an  equation  in  one  unknown  is  the 
degree  of  the  highest  power  of  the  unknown  number. 

5a; + 7  =  2x — a  is  an  equation  of  the  first  degree. 
x—b  =  4a:2  —  3  is  an  equation  of  the  second  degree, 

149.  A  simple  equation,  or  linear  equation,  is  an  equation 
which,  when  cleared  and  simplified,  is  of  the  ^rs^  degree. 

Whether  or  not  a  fractional  equation  is  a  simple  equation  cannot 
be  determined  until  it  is  cleared  of  fractions  and  the  resulting  equation 
reduced  to  its  simplest  form. 

Also,  rc2  +  x-4  =  x2  +  3  and  2x^-]-x+5  =  x'^+x{x+2)  are 
simple,  or  linear  equations. 

These  are  simple  equations,  because  when  similar  terms  are  united, 
the  square  of  the  unknown  number  disappears. 

150.  Checking  or  verifying  a  root  of  an  equation  is  the 
process  of  proving  that  the  root  satisfies  the  equation. 

This  is  done  by  substituting  the  root  found  in  the  equation 
and  ascertaining  whether  the  result  is  an  identity. 

100 


SIMPLE   EQUATIONS  101 

Solving  the  equation,  5a;  — 6+3a;+7  =  6a;+19,  we  find  the 
root  of  the  equation  to  be  9.     Substituting, 

45-6+27+7  =  54+19 

73  =  73,  an  identity 

In  checking  or  verifying  the  root  of  an  equation,  the  substitution 
should  always  be  made  in  the  original  equation. 

When  any  term  with  the  same  sign  is  found  in  both  members  of  an 
equation,  it  may  by  the  subtraction  axiom,  §  15,  be  dropped  from 
both.     Thus, 

2a:2+5a:-4  =  2a;2+2x+8 

151.  The  directions  for  solving  equations  are  generally 
summarized  in  a  rule  similar  to  the  following: 

1.  Clear  the  equation  of  fractions  by  multiplying  both  members 
by  the  lowest  common  denominator  (I.  c.  d.). 

2.  Transpose  all  unknown  terms  to  the  first  member  and  all 
known  terms  to  the  second  member. 

3.  Unite  all  terms  containing  the  unknown  number  into  one 
term,  and  unite  similar  terms  in  the  second  member. 

4.  Divide  both  members  of  the  equation  by  the  coefficient  of 
the  term  that  contains  the  unknown  number. 

Exercise  60 
Solve  and  verify  these  equations: 

1.  3(a:+l)(8x-4)  =  (6a:-2)(4x+2) 

2.  2(4a;-3)(2a:+2)  =  (4a:+7)(4x-4) 

3.  5(2a:+3)(3a;-4)  =  (6x-3)(5a;-9) 

4.  5(a:+3)+4(8-a:)  =  15a:-6(8a;+3) 
6.  (a:-2)(9-x)-(a:+5)(2-x)-7  =  0 

6.  (a:-3)(3+a:)-(4+a:)(a;-4)-2a;  =  0 

7.  4(2a:-l)-2(2a:-6)+2(a:-8)-9  =  7 


102  ELEMENTARY   ALGEBRA 

8.  5(a;+3)-4(x-2)-3(2+a:)-7  =  0 

9.  5x-3(x-6)  =  2(8-x)-4(9+x)+6 


4+^     ^8         5-  ^^-^ 8~    "^     ^3~4 


12.  43-x+-g-  =  -  13.  --x-3H-3^ 

In  checking  or  verifying  the  solution  of  a  problem,  the  substitution 
should  be  made  in  the  problem  itself. 

14.  A  has  twice  as  much  money  as  B,  and  B  has  twice  as 
much  as  C.     If  all  have  S595,  how  much  has  C? 

15.  The  sum  of  the  third,  fourth,  and  eighth  parts  of  a 
number  is  68.     Find  the  number. 

16.  The  length  of  a  rectangle  is  twice  its  width,  and  the 
perimeter  is  144  feet.     Find  the  dimensions. 

17.  A  man  gave  $125  to  his  5  sons,  each  of  4  of  them 
receiving  S5  more  than  his  next  younger  brother.  How 
much  did  the  oldest  son  receive? 

18.  James  has  J  as  many  marbles  as  Frank.  If  James  buys 
120  and  Frank  loses  23,  James  will  then  have  7  more  than 
Frank.     How  many  has  each? 

19.  The  sum  of  two  numbers  is  85,  and  3  times  the  smaller 
exceeds  twice  the  larger  by  20.     Find  the  larger  number. 

20.  Seven  men  agreed  to  share  equally  in  buying  a  boat, 
but,  as  3  of  them  were  unable  to  pay,  each  of  the  others  had 
to  pay  $30  more  than  his  original  share.  Find  the  cost 
of  the  boat. 

21.  Three  men  invested  $9400  in  business.  A  put  in  $600 
more  than  B,  and  C  invested  $200  less  than  A.  How  much 
did  A  and  C  together  invest  in  the  business? 

22.  A  farmer  sold  30  lambs  and  60  sheep  for  $300.  He 
received  twice  as  much  per  head  for  the  sheep  as  for  the 
lambs.     How  much  did  he  receive  for  the  60  sheep? 


SIMPLE  EQUATIONS  103 

152.  In  clearing  an  equation  of  fractions,  if  a  fraction  is 
preceded  by  the  minus  sign,  the  sign  of  each  term  of  the 
numerator  must  he  changed,  for  the  fraction-line  is  a  vinculum 
for  the  numerator, 

gives,  12x-8a;-96  =  6-3a; 


Exercise  61  —  Equations  and  Problems  in  One  Unknown 
Solve  the  following,  checking  some  of  them : 
3±2x     x-S     ^_4x-\-5     x-5 

„    2a:+2     2rc-3  ,  ^i     x        ,  4.T-3 

^    x-H  ,  ^       5x-12     2x-6  ,  2x-{ 

3-— +2^—4-=-^  +  ^- 

5a;+15_2a-5_^2_3a:+8_2a:-7 
*•       2         ^~    ^^~~3~      "6^ 

«.^_5(^+2(x-3)  =  6f+?^ 

.    31+8     „2     2(x-3)_4(x-2)     3(3:+2) 

6.  -^ H 2— 

7.  3^-7H2(x+3)=^+3(W) 

8.  Nine  boys  and  16  men  earn  $365  a  week.  If  each  man 
earns  4  times  as  much  as  each  boy,  how  much  do  the  9 
boys  earn  per  week? 

9.  A  boy  has  $3.60  in  dimes  and  5-cent  pieces,  and  he  has 
4  times  as  many  5-cent  pieces  as  dimes.  How  many  coins  has 
he  and  what  is  the  value  of  each  kind? 


104  ELEMENTARY  ALGEBRA 

10.  A  walked  95  miles  in  3  days,  going  4  miles  more  the 
second  day  than  the  first  and  3  miles  more  the  third  day  than 
the  second.     How  far  did  he  go  the  third  day? 

11.  A  is  3  times  as  old  as  B.  Ten  years  ago  A  was  5  times 
as  old  as  B.     Find  A's  age  now. 

Let  x  =  the  number  of  years  in  B's  age  now. 
and     3a:  =  the  number  of  years  in  A's  age  now. 

X  — 10  =  the  number  of  years  in  B's  age  10  years  ago, 
3x  — 10  =  the  number  of  years  in  A's  age  10  years  ago. 

5>{x-10)=3x-lO 

12.  A  is  4  times  as  old  as  his  son,  and  5  years  ago  he  was  7 
times  as  old.     Find  the  father's  age. 

13.  A  man  is  24  years  older  than  his  son.  Fourteen  years 
ago  he  was  3  times  as  old.     Find  the  age  of  each. 

14.  A  farmer  sold  corn,  wheat,  and  oats.  For  his  corn  and 
wheat  he  received  $800.  For  his  corn  and  oats  he  received 
$720,  and  for  his  wheat  and  oats  $840.  How  much  did  he 
receive  for  all  his  grain? 

16.  A  man  spent  J  of  his  money  for  a  suit  of  clothes, 
J  of  it  for  a  watch,  and  had  $115  left.  How  much  did  he 
spend? 

16.  The  sum  of  two  numbers  is  82,  and  if  the  greater  is 
divided  by  the  less,  the  quotient  is  5  and  the  remainder  4. 
Find  the  two  numbers. 

17.  D  is  6  years  older  than  C;  C  is  4  years  older  than  B; 
B  is  3  years  older  than  A.  If  they  live  5  years,  the  sum  of 
their  ages  will  be  135  years.     Find  D's  age. 

18.  A  grocer  mixed  tea  worth  70^  a  pound  with  tea  worth 
50^  a  pound  in  such  proportions  that  the  mixture  weighing 
100  pounds  was  worth  $58.  How  many  pounds  of  each  kind 
were  in  the  mixture? 


SIMPLE  EQUATIONS  105 

19.  A  man  paid  a  bill  of  $12.95  in  quarters,  dimes,  and 
5-cent  pieces,  giving  in  payment  3  times  as  many  dimes  as 
5-cent  pieces,  and  twice  as  many  quarters  as  dimes.  How 
many  coins  were  there  in  the  whole  amount? 

20.  Find  two  numbers  differing  by  96,  the  sum  of  which  is 
equal  to  twice  their  difference. 

.     21.  Divide  32  into  two  parts  such  that  the  sum  of  twice  the 
less  and  5  times  the  greater  shall  be  118. 

22.  A  man  divided  $3500  among  his  5  sons  so  that  each 
one  received  $100  more  than  his  next  younger  brother. 
How  much  did  the  youngest  son  receive? 

23.  A  farmer  sold  |^  of  his  potatoes  and  had  left  665  bushels 
less  than  he  sold.  Find  the  value  of  his  whole  crop  at  55^  a 
bushel. 

24.  The  sum  of  three  numbers  is  170.  The  second  exceeds 
the  first  by  8,  and  the  third  is  14  less  than  the  second.  Find 
the  sum  of  the  second  and  third  numbers. 

26.  A  lady  bought  a  hat  and  a  dress  for  $72,  and  the  differ- 
ence in  the  cost  was  4  times  the  cost  of  the  hat.  How  much 
did  she  pay  for  the  dress? 

26.  A  man  bequeathed  his  property,  which  amounted  to 
$30,300,  to  his  wife,  son,  and  daughter.  The  son  received 
$1200  more  than  the  daughter  and  $3000  less  than  the  wife. 
How  much  did  the  wife  receive? 

27.  A  man  paid  $12,800  for  two  houses  and  a  farm,  paying 
the  same  sum  for  each  house.  If  he  had  paid  twice  as  much 
for  each  house,  the  two  houses  would  have  cost  $1600  more 
than  the  farm.     Find  the  cost  of  the  farm. 

28.  There  are  3  times  as  many  pupils  in  one  school  as  in 
another.  If  120  pupils  were  taken  from  the  larger  school  to 
the  smaller,  the  larger  would  still  have  twice  as  many  as 
the  smaller.     How  many  are  there  in  both  schools? 


106  ELEMENTARY   ALGEBRA 

Exercise  62  —  Problems  in  Simultaneous  Equations 

Solve  the  following  problems  in  simultaneous  simple 
equations : 

1.  The  sum  of  two  numbers  is  85,  and  their  difference 
exceeds  J  of  the  smaller  by  8.     Find  the  numbers. 

Let  X  =  the  larger  number, 
and  i/  =  the  smaller  number. 

y 

x-y-8=- 
5 

The  second  equation  contains  a  fraction.  Clear  this  of  fractions 
and  then  with  the  other  equation,  ehminate. 

2.  If  5  is  added  to  the  numerator  of  a  certain  fraction,  its 
value  is  f ;  and  if  1  is  subtracted  from  the  denominator,  its 
value  is  J.     Find  the  fraction. 

Let  n  =  the  numerator, 
and  d  =  the  denominator. 

3.  Three  times  the  larger  of  two  numbers  exceeds  ^  of  the 
smaller  by  66,  and  3  times  the  smaller  exceeds  ^  of  the 
larger  by  46.     Find  the  numbers. 

4.  If  3  is  added  to  both  terms  of  a  certain  fraction,  its 
value  is  f ;  and  if  4  is  subtracted  from  both  terms,  its  value 
is  f .     Find  the  fraction. 

5.  A  miller  bought  50  bushels  of  corn  and  40  bushels  of 
oats  for  $64.  At  another  time  he  bought  at  the  same  prices 
38  bushels  of  oats  and  70  bushels  of  corn  for  $78.80.  How 
much  did  he  pay  for  all  of  the  corn? 

6.  A  dealer  bought  oranges,  some  at  2  for  5^  and  some 
at  3  for  5^,  paying  $12  for  all.  Three  dozen  were  unsalable, 
but  he  sold  the  remainder  at  30^  a  dozen,  making  a  profit 
of  $2.10.     How  many  oranges  did  he  buy? 


CHAPTER  XI 
DIVISION 

153.  Division  is  the  process  of  finding  one  of  two  numbers 
when  their  product  and  the  other  number  are  known. 

154.  The  dividend  is  the  number  to  be  divided  and  repre- 
sents the  product  of  the  two  numbers. 

155.  The  divisor  is  the  number  by  which  we  divide  and 
represents  one  factor  of  the  dividend. 

156.  The  quotient  is  the  number  obtained  by  division  and 
represents  the  other  factor  of  the  dividend. 

Since  division  is  the  reverse  of  multiplication,  the  rule  for  division  is 
derived  from  the  process  of  multiplication. 

Three  things  must  be  determined :   The  sign  of  the  quotient, 
the  coefficient  J  the  exponent  of  each  letter. 


DIVIDING   A   MONOMIAL  BY   A   MONOMIAL 

157.  The  Sign  of  the  Quotient. 

(+7)(+5)  =  +35,  therefore  (+35)^(+5)  = +7 

(+7)(-5)  =  -35,  therefore  (-35)^(-5)  = -f 7 

(-7)(-5)  =  +35,  therefore  (+35)-^(-5)  = -7 

(-7)(+5)  =  -35,  therefore  (-35)^(+5)  = -7 

158.  Sign  Law  of  Division. —  Like  signs  of  dividend  and 
divisor  give  a  positive  quotient;  unlike  signs,  a  negative  quotient. 

107 


108  ELEMENTARY  ALGEBRA 

Give  the  following  quotients : 

(+56)-^(+7)     (-64)-(-8)     (96)-f-(-8)     (-63)^(+9) 
(+84)^(-7)     (-72)-(-9)     (68)-^(-4)     (+75)-^(-5) 
Since  5aX3x=loax,  therefore  15ax-^3x  =  5a 
The  coefficient  of  the  quotient  is  the  coefficient  of  the  dividend 
divided  by  the  coefficient  of  the  divisor. 

159.  The  Exponent  in  the  Quotient.  Since  the  dividend 
is  a  product,  one  factor  of  which  is  the  divisor,  the  exponent 
of  the  dividend  is  the  sum  of  the  exponents  of  divisor  and 
quotient. 

To  find  the  exponent  of  the  quotient,  subtract  the  exponent 
of  the  divisor  from  that  of  the  dividend. 

160.  Law  of  Exponents  for  Division. —  Each  exponent  in 
the  divisor  is  subtracted  from  the  exponent  of  the  same  letter  in 
the  dividend. 

Since  a^Xa'^  =  a^,  therefore  a^-^a^  =  a^ 

In  general  numbers, 

Observe  the  following: 

2ab)W^  -2a^b)-M¥  -Za^h)l2a^¥c 

4a^b  Aa'^b'^  —  4a6c 

By  the  law  of  exponents  for  division,  a^-i-a^  =  a^.  But  any 
number,  except  0,  divided  by  itself  also  equals  1.    Therefore, 

a«  =  l. 

161.  Meaning  of  Exponent  0.  Since  a  may  represent  any 
number,  it  follows  that  any  number  with  a  zero-exponent  is 
equal  to  1.     Thus, 

2abcP  =  2ab-l  =  2ab 

From  this  equation  it  is  evident  that  any  letter  with  a  zero-expo- 
nent may  be  omitted  from  a  term,  because  its  presence  only  multiplies 
the  rest  of  the  term  by  1. 


DIVISION  109 

Exercise  53  —  Dividing  Monomials 

Find  the  following  quotients : 

1.  46c3)166V  2.  Qx^y)-SOxY  3.  -xyz)-2x^yz^ 

4.  Qxy'')lSx^y^  5.  7a'x)-2Sa''x^  6.  -acx)-5ac^x^ 

7.  Scx^)l5cH^  8.  9b^c)-lS¥(^  9.  -hxy)-db^xy^ 

10.  962)18a53c2  11.  3a:)-21a2xy  12.  -axy)-4axY 

13.  2ac3)14a3c3  14.  8a;?/)-24xy  15.  -bcx)-Wcx^ 

16.  8xi/^)16a^Y  17.  Qa)-lSa^¥c''  18.  -  a6a^)  -  Sab^a;^ 

19.  5?/2)15a;?/322  20.  5a^x)-S5a^x'^  21.  -xyz)-7x^yz^ 

DIVIDING   A   POLYNOMIAL  BY   A    MONOMIAL 

162.  Since  division  is  the  inverse  of  multiplication,  (see 
§§  142,  143),  to  divide  a  polynomial  by  a  monomial,  we 
divide  each  term  of  the  polynomial  by  the  monomial  divisor. 
Thus, 

5a62  c)20a''¥c-15a^b^c^+25ab^(^ 

4a62    -  Sa^c    +  Sc^ 

Exercise  54  —  Dividing  a.  Polynomial  by  a  Monomial 

Divide : 

1.  32a63c2-l66Vc?+1663c*-864c2  by  Wc'' 

2.  14a6*c2+286V(^-186V+26V  by  2b^c^ 

3.  10b^x^y-25¥cx^+15b*x^-5¥x^  by  5¥x^ 

4.  15aV-12a36c2(i+18aV-3a4c2  by  Sa'c^ 

5.  18a;V2+24axy-12x4i/3+6a:y  by  Ga^V 

6.  36x^3+ 18xV- 27 6a;4!/3z+9a:Y  by  9x¥ 

7.  16a^a:3-24a36cx4+32a3a;3-8a2a:^  by  Sa^x^ 

8.  21a363c+14a464rf-28a26^-7a263  by.  7a^b^ 


no  ELEMENTARY   ALGEBRA 

DIVIDING   A   POLYNOMIAL  BY   A   POLYNOMIAL 

163.  The  rule  for  dividing  a  polynomial  by  a  polynomial 
is  deduced  from  the  process  of  multiplication. 

Study  this  example  carefully: 

.    a4+2a36-6a2fe2+26a63-156^|a2+4afe-362 
a^+4a^6-3a262  a'-2ab-\-5b^ 

-2a36-3a262+26a63  ' 

5a262+20a63-1564 
ba'b^  +  20ab^-15b'^ 

Arrange  the  dividend  and  divisor  with  reference  to  the  descending 
powers  of  a,  writing  the  divisor  at  the  right  of  the  dividend. 

Since  the  dividend  is  the  product  of  the  divisor  and  quotient,  it  is 
the  algebraic  sum  of  the  products  obtained  by  multiplying  the  divisor 
by  the  several  terms  of  the  quotient. 

Hence,  when  dividend,  divisor,  and  quotient  are  arranged  with 
reference  to  the  descending  powers  of  some  letter,  the  first  term  of  the 
dividend  is  the  product  of  the  first  terms  of  the  divisor  and  quotient, 
whence  the  first  term  of  the  quotient  is  the  quotient  of  the  first  term  of 
the  dividend  divided  by  the  first  term  of  the  divisor. 

Dividing  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  we  have  a^  for  the  first  term  of  the  quotient. 

Since  the  dividend  is  the  algebraic  sum  of  the  products  obtained 
by  multiplying  the  divisor  by  the  several  terms  of  the  quotient,  if  the 
product  of  the  divisor  and  first  term  of  the  quotient  is  subtracted  from 
the  dividend,  the  remainder,  which  is  a  new  dividend,  is  the  product 
of  the  divisor  and  the  other  terms  of  the  quotient,  and  the  next  term 
of  the  quotient  is  the  quotient  of  the  first  term  of  the  remainder  divided 
by  the  first  term  of  the  divisor. 

Dividing  the  first  term  of  the  remainder  by  the  first  term  of  the 
divisor,  we  have  —2ab  for  the  second  term  of  the  quotient. 

Repeating  this  process  until  there  is  no  remainder,  we  obtain  the 
quotient  a^  —  2ab-\-5b^. 

Each  remainder  must  be  arranged  in  the  same  manner  as  the  divi- 
dend and  divisor. 


DIVISION  111 

Observe  the  following  solutions: 
(I)  a4-a262+2a63  -     b^     \a^-ab+b^ 

a^-a'b  +  d'b'  a^^ab-b^ 

a^b  -2a262+2a63 

a^b  -  a^b^^  a¥ 

-  a262+  ab^-¥ 

-  a^62+  ab^-b^ 

(II).         x'+4y'  \x'-2xy+2y' 


Divide: 


x'-2x'y+2x^y^  x^+2xy-^2y^ 

2x^y-2xY+^y'' 
2x^y — 4x'^y'^ + 4:xy^ 

2xhf-4xy^+4:y* 


Exercise  65  —  Dividing  Polynomials 

1.  a2-a-42by  a+6 

2.  x^-x-SO  by  x-6 

3.  a2+a- 72  by  a+9 

4.  x^-\-x-5Qhy  x-7 
6.  a2-6a-16by  a+2 

6.  a:2+8x-33by  a-3 

7.  a2-9a-52by  a+4 

8.  x2+7a:-98by  a;-7 

9.  a2+32a+60  by  a+2 

10.  a:2-17a;-18by  l-\-x 

11.  a2+25a-54by  a-2 


112  ELEMENTARY   ALGEBRA 

12.  a;2+20x-f75bya;+15 

13.  a*-15a2+56by  a2-7 

14.  3a6-8a3-28by  a3+2 

15.  5x8+42x^+85  by  x*+5 

16.  a^— a(^  —  3a^c-\-c^  by  a  — c 

17.  ¥-\-b^x  —  hx^  —  oir^  by  h  —  x 

18.  a^+ax^+a^x+o^  by  a+x 

19.  a4+64+a2fe2bya2+fe2_a6 

20.  ax''^+G6x+ 6x^+62  by  ax+?> 

21.  x^+3x?/^+3x2|/+i/^  by  x+i/ 

22.  x^  — 2/^+2;^  — 2x2;  by  X  — 2/  — 2; 

23.  a^-\-2xy  —  y^  —  x'^  by  a+x  —  ?/ 

24.  32x3-6x-l  by  8x2-2x-l 

25.  4a4+6a2+8a3-24  by  2a+4 

26.  a3+27  by  a+3  27.  36x2-812/2  by  6x+9i/ 

28.  a^-ie  by  a-2  29.  27x3+642/^  by  3x+42/ 

30.  a3-64bya-4  31.  25x2-162/2  by  5x-4?/ 

32.  0^-81  by  a+3  33.  Sx^-125y^  by  2x-5y 

34.  a6+27bya2+3  35.  16x2-64^/2  by  4x+Sy^ 

36.  a9-64  by  a3-4  37.  IQx'-Sly*  by  2x+3t7 

38.  a8-16  by  a2+2  39.  25x^-49y^  by  5x-7y 


CHAPTER  XII 

APPLICATIONS  OF  SIMPLE  EQUATIONS. 
ELIMINATION  BY  SUBSTITUTION 

SUGGESTIONS  ON  PROBLEM-SOLVING 

164.  Read  the  following  suggestions  carefully: 

I.  Solving  problems  in  algebra  is  finding  one  or  more 
unknown  numbers  by  the  use  of  equations. 

II.  Bear  in  mind  that  in  all  problem-solving  the  general 
plan  is  to  find  two  different  expressions  to  represent  the 
same  number,  place  them  equal  to  form  an  equation,  and 
solve  the  equation. 

III.  The  absolutely  necessary  condition  of  success  in 
this  work  is  the  power  to  focus  every  faculty  of  the  mind 
on  the  task  in  hand.  For  the  time  being  every  thought  of 
other  things  must  he  banished  from  the  mind: 

IV.  First  of  all,  you  should  read  the  problem  attentively 
and  thoughtfully  several  times  before  you  attempt  to  form 
the  equation.  The  purpose  of  this  careful  reading  is  to  see 
clearly  what  facts  are  given  and  what  is  to  be  found. 

V.  Very  few  advanced  pupils  can  see  through  a  problem  at 
a  glance  and  determine  the  equation,  and  of  course  a  beginner 
cannot  do  it.  You  must  not  allow  this  partial  and  hazy 
grasp  of  the  problem  to  discourage  you.  Never  permit 
yourself  even  to  think  that  you  cannot  conquer  the  problem. 

VI.  You  must  advance  by  short  steps  at  first.  Here  is  a 
most  important  suggestion  for  you:  Do  not  at  the  outset 
try  to  see  every  number  in  the  problem  represented  in  sym- 
bols and  even  to  see  the  equation  to  be  used,  for  very  few 

113 


114  ELEMENTARY  ALGEBRA 

can  do  that;  but  express  in  symbols  all  you  can  of  the  con- 
ditions of  the  problem,  no  matter  how  useless  this  may  seem. 
From  these  expressions  you  will  see  what  numbers  are  equal, 
and  the  formation  of  the  equation  will  become  a  simple 
matter. 

VII.  It  is  much  easier  to  reason  about  small  numbers  than 
about  large  ones.  If  the  numbers  in  a  problem  are  large, 
or  complicated,  or  are  general  numbers,  simplify  the  problem 
by  replacing  them  with  simple  arithmetical  numbers;  then 
reread  the  problem  using  the  simple  numbers,  and  try  again 
to  sense  the  meaning.  To  form  the  habit  of  doing  this  will 
help  you  greatly. 

VIII.  School  work  that  requires  little  or  no  effort  on  your 
part  will  not  increase  your  power  to  do  harder  things.  You 
should  welcome  some  tasks  that  test  you  to  the  limit;  and  if 
you  would  grow  stronger,  you  must  always  rely  upon  yourself. 
It  is  ruinous  to  your  progress  to  rely  on  others  to  assist  you 
in  solving  your  problems. 

IX.  Appeal  to  your  teacher  for  assistance  only  after  you 
have  really  done  your  best,  and  then  ask  only  for  one  or  two 
hints  to  start  you  right. 

Exercise  56  —  Problems  Requiring  Simple  Equations 

Solve  the  following  equations  and  problems: 

^    x+5    x-\-2.x-d_.  ^    s+3,  s+1     s+8_. 

^*  "3  6~+^       ^  ^*  ~r"+"3  5~~^ 

,    7n+4  ,  „       3n-9     „,  ^     .       5x-S     ,„     8:r+5 

3.  ^-tan— 3-  =  31  4.  40.-^—17  =  ^- 

^    22/-5  ,  35     _       62/-h3               ^    n+9    n+5_       n+9 
6.  -6-+^-52/         ^  6.  — __n— ^ 

^    S+5.S-1  s+7 


APPLICATIONS  OF  SIMPLE  EQUATIONS  115 

8.  The  sum  of  two  numbers  is  94,  and  their  difference  is 
38.     Find  the  numbers. 

9.  A  boy  has  3  times  as  many  dimes  as  quarters,  and  he 
has  $11  in  all.     How  many  coins  has  he? 

10.  Seven  times  a  certain  number  is  176  more  than  3 
times  the  number.     Find  the  number. 

11.  A  man  bought  50  sheep,  some  at  $3.75  a  head  and  the 
others  at  $4.50  a  head.  The  average  cost  was  $4.05.  How 
many  did  he  buy  at  the  lower  price? 

12.  A  boy  earns  $1.25  a  day  less  than  his  father,  and  in  14 
days  the  father  earns  $15  more  than  the  son  earns  in  16  days. 
How  much  do  both  earn  per  day? 

13.  A  clerk  spends  J  of  his  annual  salary  for  board,  | 
for  clothes,  J  for  other  expenses,  and  saves  $1100.  How 
much  are  his  annual  expenses? 

14.  At  what  rate  per  annum  will  $8000  yield  $540  interest 
in  1  year  and  6  months? 

Let  X  =  the  rate  per  annum. 
8000  Xj^x|  =  540 

16.  A  man  invested  a  certain  sum  at  5%  and  twice  as 
much  at  6%.  His  annual  income  from  both  investments 
was  $680.     How  much  did  he  invest? 

16.  A  is  64  years  old,  and  B  is  f  as  old.  How  many  years 
have  passed  since  B  was  J  as  old  as  A? 

17.  The  sum  of  two  numbers  is  84,  and  7  times  the  less 
exceeds  5  times  the  greater  by  12.     Find  the  numbers. 

18.  A  had  8  acres  of  land  less  than  B,  but  A  sold  24  acres 
to  B.  A  then  had  left  only  J  as  many  acres  as  B.  How  many 
acres  did  each  have  at  first? 


116  ELEMENTARY   ALGEBRA 

19.  A  woman  bought  36  yards  of  silk  for  $31,  paying  75^^  a 
yard  for  part  of  it  and  $1  a  yard  for  the  rest.  How  rnan}^ 
yards  of  each  kind  did  she  buy? 

20.  A  grocer  has  tea  worth  40^  a  pound  and  some  worth 
GOjzf  a  pound.  How  many  pounds  of  each  must  he  take  to 
mix  60  pounds  worth  54^  a  pound? 

Solve  the  20th  with  one  and  then  with  two  unknown  numbers. 

21.  A  boy  bought  a  number  of  apples  at  the  rate  of  7  for  10|^ 
and  sold  them  at  the  rate  of  10^  for  3,  gaining  $2.  How  many 
apples  did  he  buy? 

22.  If  it  costs  the  same  at  $1  a  yard  to  enclose  a  square 
court  with  a  fence  as  to  pave  it  at  10^  a  square  yard,  what 
are  the  dimensions  of  the  court? 

23.  A  mason  received  $3.60  a  day  for  his  labor  and  paid 
85^  a  day  for  his  board.  At  the  end  of  44  days  he  had 
saved  $92.20.     How  many  days  did  he  work? 

24.  A,  B,  and  C  together  earn  $5000.  A's  salary  is  f  of 
B's  and  $450  less  than  C's.     Find  C's  salary. 

26.  The  sum  of  J  and  J  of  a  number  exceeds  5  times  the 
difference  between  ^  and  ^  of  the  number  by  29.  Find 
the  number. 

26.  If  f  of  a  certain  principal  is  invested  at  4%  and  the 
remainder  at  5%,  the  annual  income  is  $690.  Find  the 
whole  sum  invested. 

27.  A  bought  sheep  at  $4  a  head  and  had  $33  left.  If  he 
had  bought  them  at  $4.75  a  head,  he  would  have  needed 
75^  more  to  pay  for  them.     How  many  did  he  buy? 

28.  The  length  of  a  rectangle  exceeds  its  width  by  13 
inches.  If  the  length  were  diminished  7  inches  and  the  width 
increased  5  inches,  the  area  would  remain  the  same.  What 
are  the  dimensions  of  the  rectangle? 


APPLICATIONS  OF  SIMPLE   EQUATIONS  117 

29.  What  is  the  distance  between  two  cities,  if  an  express 
train  which  runs  60  miles  an  hour  can  go  from  one  city  to 
the  other  in  6  hours  less  time  than  a  freight  train  which 
runs  20  miles  an  hour? 

30.  A  man  owed  $140.  He  sold  wheat  at  $1  a  bushel  and 
corn  at  75^  a  bushel,  selUng  the  same  number  of  bushels  of 
each,  and  received  just  money  enough  to  pay  the  debt. 
How  many  bushels  of  grain  did  he  sell? 

31.  A  man  was  employed  for  56  days  at  the  rate  of  $3.25  a 
day  and  his  board,  and  for  every  day  he  might  be  idle  he  was 
to  pay  $1  for  his  board.  At  the  end  of  the  time  he  received 
$148.     How  many  days  did  he  work? 

32.  Two  wheelmen  are  144  miles  apart.  They  ride  toward 
each  other,  A  riding  8  miles  an  hour  and  B  6  miles  an  hour. 
B  sets  out  3  hours  before  A.  How  many  miles  w411  A  have 
I'idden  when  they  meet? 

33.  At  what  time  between  3  and  4  o'clock  are  the  hands  of  a 
clock  together? 

Let  m  =  the  number  of  minute-spaces 
passed  over  by  the  minute-hand  before  the 
hands  are  together. 

Since  the  minute-hand  goes  12  times  as 
fast  as  the  hour-hand,  m  divided  by  12  = 
the  number  of  minute-spaces  passed  over  by 
the  hour-hand  in  the  same  time. 

The  hour-hand  must  pass  over  15  spaces 
from  12  to  3,  and,  in  addition,  as  many  as  the  hour-hand  passes  over  in 
the  meantime.     Hence  the  equation  is 

m 

Solving,  the  number  of  spaces  passed  over  by  the  minute- 
hand  is  16^^,  and  the  time  is  16yy  minutes  past  3. 


118 


ELEMENTARY   ALGEBRA 


34.  At  what  times  between  5  and  6  o'clock  are  the  hands  of 
a  clock  at  right  angles  to  each  other? 


The  hands  are  at  right  angles  twice  between  5  and  6,  once  before  the 
minute-hand  passes  the  hour-hand,  and  once  after. 

In  the  first  case,  the  minute-hand  must  pass  over  25  spaces,  pkis  the 
number  of  spaces  passed  over  by  the  hour-hand,  minus  15  spaces. 

In  the  second  case,  the  minute-hand  must  pass  over  25  spaces,  plus 
m  divided  by  12,  plus  15  spaces.     The  two  equations  are 


m  =  25H -15 

12 


m  =  25+-  +  15 


36.  At  what  time  between  3  and  4  o'clock  are  the  hands  of  a 
clock  opposite  «ach  other? 

m 
m  =  15  + -4-30 
1  z 

36.  At  what  time  between  8  and  9 
o'clock  are  the  hands  of  a  clock  to- 
p;ether? 

37.  At  what  time  between  2  and  3 
o'clock  are  the  hands  of  a  clock  at  right  angles  to  each  other? 

38.  A  is  54  years  old,  and  B  is  \  as  old.     In  how  many- 
years  will  B  be  \  as  old  as  A? 

39.  A,  B,  and  C  together  earn  $3650.     A's  salary  is  J  of 
B's  and  $650  less  than  C's.     Find  C's  salary. 

40.  A  boy  has  $11  in  half-dollars  and  5-cent  pieces,  in  all 
58  coins.     How  many  has  he  of  each  kind? 


APPLICATIONS  OF  SIMPLE  EQUATIONS  119 

41.  Find  the  number  whose  double  dhninished  by  23  is  as 
much  greater  than  53  as  68  is  greater  than  the  number. 

42.  A  is  28  years  older  than  his  son,  but  5  years  ago  he 
was  3  times  as  old.     Find  the  father's  age. 

43.  A  man  bought  some  cows  at  $40  a  head.  If  he  had 
bought  2  less  for  the  same  money,  each  would  have  cost  $10 
more.     How  many  did  he  buy? 

44.  A  had  twice  as  many  sheep  as  B.  Each  sold  half  his 
flock  to  C,  and  A  sold  30  to  B,  whereupon  A  and  B  had  the 
same  number.     How  many  had  each  at  first? 

46.  One  of  two  numbers  is  4  times  the  other.  If  24  is  sub- 
tracted from  the  greater,  and  the  less  is  subtracted  from  66, 
the  remainders  are  equal.     Find  the  numbers. 

46.  A  woman  bought  12  yards  of  silk,  but  if  she  had  bought 

8  yards  more  for  the  same  money,  it  would  have  cost  60<;^ 
a  yard  less.     How  much  did  it  cost? 

47.  A  father  and  two  sons  earn  $222  a  month,  the  two  sons 
receiving  the  same  wages.  If  the  sons'  wages  were  doubled, 
they  would  together  receive  only  $6  less  than  their  father. 
How  much  does  the  father  earn  per  month? 

48.  A  man  bought  land  at  $90  an  acre  and  had  $1000 
left.  At  $105  an  acre,  he  would  have  needed  $200  more  to 
pay  for  it.     How  many  acres  did  he  buy? 

49.  A  fruit  dealer  bought  some  oranges  at  the  rate  of  3  for 
5^  and  twice  as  many  others  at  the  rate  of  2  for  5fj.  He  sold 
them  all  at  36cf  a  dozen  and  made  a  profit  of  $5.60.  How 
many  oranges  did  he  buy? 

50.  A  pedestrian  walked  a  certain  distance  at  the  rate  of  if 
miles  an  hour.  He  rested  2  hours  at  the  end  of  his  journey 
and  returned  at  the  rate  of  2^  miles  an  hour.     If  he  was  out 

9  hours,  how  many  miles  did  he  walk? 


120 


ELEMENTARY   ALGEBRA 
ELIMINATION   BY   SUBSTITUTION 


165.  The   following   example  illustrates  the   method   of 
elimination  by  substitution: 

Sx+2y  =  Q5 


Transposing  2?/  in  (1), 
Dividing  (3)  by  3, 

Substituting  in  (2), 


4x-Sy  =  S0 
Sx  =  Q5-2y 
Q5-2y 


x  = 


260-8?/ 


-dy  =  SO 


(1) 
(2) 

(3) 
(4) 

(5) 


Solving  (5),  we  have  the  value  of  y,  and  substituting  this 
value  in  (1)  or  (2),  we  find  the  value  of  x. 

166.  Rule. —  Determine  first  which  of  the  two  unknown  num- 
bers it  is  more  convenient  to  eliminate. 

From  either  equation,  find  the  valu^  of  that  unknown  number 
in  terms  of  the  other.  Substitute  this  value  for  the  same  un- 
known number  in  the  other  equation. 


Exercise  57 


Eliminate  by  substitution  and  solve: 


1. 


3. 


7. 


f  4x  —  6?/  =  6 
\2a;+3i/  =  9 

3a;-3i/  =  9 
4a;- 52/ =  7 

f4x+2?/  =  5 
\5a;+3^  =  7 

bx    ^y 
4^8 


f5x+4!/=  —4 

\4x+37/=-2 


4. 


6. 


8. 


4a;-5?/=-2 
3a:-4i/=-3 

f  2x4-2!/ =-5 
\6a;+9?/=-6 


3^_% 

y  y 

9x 
14 


3i/=- 


ELIMINATION   BY  SUBSTITUTION  121 

Exercise  58  —  Problems — Eliminate  by  Substitution 

1.  The  sum  of  the  two  digits  that  express  a  number  is  14; 
and  if  18  is  added  to  the  number,  the  digits  are  interchanged. 
Find  the  number. 

Let  i  =  the  digit  in  tens'  place, 

and  u  =  the  digit  in  units'  place. 

t-\-u  =  U 

iot+ii+m  =  iOu-\-t 

2.  The  sum  of  the  two  digits  of  a  number  is  12;  and  if  the 
number  is  divided  by  the  sum  of  the  digits,  the  quotient  is  7. 
Find  the  number. 

3.  The  sum  of  the  two  digits  of  a  number  is  12;  and  if  18 
is  subtracted  from  the  number,  the  digits  are  interchanged. 
Find  the  number. 

4.  In  6  hours  A  rides  9  miles  more  than  B  does  in  5  hours, 
and  in  10  hours  B  rides  2  miles  more  than  A  does  in  7  hours. 
How  manj^  miles  does  each  ride  per  houi? 

6.  The  sum  of  the  two  digits  of  a  number  is  14;  and  if  the 
digits  are  interchanged,  the  resulting  number  exceeds  the 
given  number  by  18.     Find  the  number. 

6.  A  number  exceeds  4  times  the  sum  of  its  two  digits  by  6. 
If  the  number  is  divided  by  the  tens'  digit,  the  quotient  is 
10  and  the  remainder  4.     Find  the  number. 

7.  A  man  invested  $28,000,  partly  in  5%  bonds  and  partly 
in  6%  bonds.  The  annual  income  from  the  5%  bonds  ex- 
ceeds the  annual  income  from  the  6%  bonds  by  $80.  How 
much  did  he  invest  at  each  rate? 

8.  A  dealer  bought  60  barrels  of  apples  and  10  barrels  of 
pears  for  $195.  He  sold  the  apples  at  a  profit  of  40%  and  the 
pears  at  a  profit  of  20%,  receiving  $264  for  all.  How  much 
per  barrel  did  he  receive  for  each  kind  of  fruit? 


122  ELEMENTARY   ALGEBRA 

9.  In  4  years  a  sum  of  money  at  simple  interest  amounts 
to  $768,  and  in  5  years  at  the  same  rate  it  amounts  to  $800. 
Find  the  sum  invested  and  the  rate. 

10.  A  pound  of  tea  and  5  pounds  of  coffee  cost  $2.  At 
prices  20%  higher,  3  pounds  of  tea  and  11  pounds  of  coffee 
would  cost  $6.     Find  the  price  of  each. 

11.  If  7  is  added  to  the  sum  of  the  two  digits  of  a  certain 
number,  the  result  is  5  times  the  tens'  digit,  and  if  45  is 
added  to  the  number  itself,  the  digits  are  interchanged. 
Find  the  number. 

12.  If  the  sum  of  two  numbers  is  divided  by  5,  the  quotient 
is  21  and  the  remainder  4;  and  if  the  difference  of  the  numbers 
is  divided  by  10,  the  quotient  is  6  and  the  remainder  3. 
Find  the  numbers. 

13.  A  man  paid  $14  for  oranges,  buying  some  of  them  at  12 
for  25^  and  the  rest  at  14  for  25^.  He  sold  them  all  at  30cf 
a  dozen  and  made  a  profit  of  $4.30.  How  many  did  he  buy 
of  each  kind? 

14.  If  the  larger  of  two  numbers  is  divided  by  the  smaller, 
the  quotient  is  6  and  the  remainder  8;  but  if  7  times  the 
smaller  is  divided  by  the  larger,  the  quotient  is  1  and  the 
remainder  9.     Find  the  numbers. 

16.  If  the  numerator  of  a  certain  fraction  is  doubled  and  3 
added  to  the  denominator,  its  value  is  f ;  if  the  denominator 
is  doubled  and  2  added  to  the  numerator,  its  value  is  y. 
Find  the  fraction. 

16.  If  a  rectangular  plot  of  land  were  20  feet  longer  and 
10  feet  wider,  the  area  would  be  increased  3000  square  feet; 
but  if  the  length  were  10  feet  more  and  the  width  30  feet 
less,  the  area  would  be  diminished  2400  square  feet.  How 
many  square  feet  are  there  in  the  plot? 


CHAPTER  XIII 

GENERAL   NUMBERS.    FORMULAS. 
TYPE-FORMS 

GENERAL  NUMBERS 

167.  Representing  Numbers.  B}^  common  usage,  the 
Arabic  numerals  of  arithmetic  and  the  letters  used  in  algebra 
are  called  numbers.  It  must  be  remembered,  however,  that 
all  number  symbols  are  used  simply  to  represent  numbers. 

Since  letters  are  used  in  algebra  to  represent  any  numbers, 
these  letters  are  called  general  numbers. 

168.  A  general  number  is  a  letter  or  other  number  symbol 
that  may  represent  any  number. 

To  be  able  to  read  algebraic  expressions  in  concise  English 
and  to  express  mathematical  statements  in  algebraic  symbols  is 
of  great  importctfice. 

For  example,  Sab,  dax,  or  Sxy  represents  three  times  the  product  of 
any  two  numbers.  Also,  2{a  —  b)  or  2{x  —  y)  may  represent  twice  the 
difference  of  any  two  numbers. 

Since  a  and  b  may  represent  any  two  unequal  numbers,  the 
equality — 

ia-\-b)-{a-b)=2b 

expresses  the  following  principle: 

The  sum  of  any  two  unequal  numbers  exceeds  their  difference 
by  twice  the  smaller  number. 

If  a  and  b  are  any  two  numbers  of  which  b  is  the  smaller, 
what  principle  does  this  equality  express 

2(a+6)-2(a-5)=46? 
What  principles  do  the  following  identities  express 
(a-f-6)  +  (a-6)=2a  ia+iy-a^  =  2a-\-l? 

123 


124  ELEMENTARY   ALGEBRA 

FORMULAS 

169.  A  formula  is  an  expression  of  a  general  principle,  or 
rule  in  general  number  symbols  and  in  the  form  of  an 
equality. 

The  expression  of  a  formula  in  words  is  a  principle,  and  the 
expression  of  it  as  a  direction  is  a  rule. 

The  ability  to  express  general  principles  as  formulas,  and 
to  read  formulas  accurately  as  principles  and  rules  is  of  the 
greatest  value  to  students  of  algebra,  physics,  etc. 

The  truth  of  the  following  algebraic  statement,  called  a 
formula,  may  be  verified  by  performing  the  indicated  oper- 
ations : 

ix+yy-{x-yy  =  4:xy 

Supposing  that  x  and  y  are  any  two  numbers,  what 
principle  does  the  formula  express? 

Exercise  59 

1.  Verify  the  truth  of  this  formula: 

{a-\-xy  —  {a-{-x)  {a  —  x)  =2x{a-\-x) 

2.  Having  verified  the  truth  of  this  algebraic  statement, 
tell  what  general  principle  it  expresses. 

Since  a  fornmla  expresses  a  general  principle,  it  applies 
to  all  particular  examples  of  that  type. 

3.  By  how  much  does  687  +  125  exceed  687-125?  By 
how  much  does  2(759+45)  exceed  2(759-45)? 

4.  How  much  does  the  square  of  50+3  exceed  the  square 
of  50  —  3?     Give  result  without  squaring. 

6.  Without  squaring  the  binomial,  give  the  difference  be- 
tween (20+6)2  and  (20+6)  (20 -6). 

6.  By  how  much  does  569+350  exceed  569  -  350?  By  how 
much  does  3(476  +  150)  exceed  3(476-150)? 


FORMULAS  125 

7.  How  much  does  the  square  of  40+5  exceed  the  square  of 
40  —  5?     Give  the  result  without  squaring. 

170.  Deriving  Formulas.  The  use  of  general  numbers 
enables  us  to  derive  formulas  for  solving  whole  classes  of 
problems. 

General  numbers  may  be  used  to  represent  any  units  of 
measure  as  well  as  to  represent  abstract  numbers. 

We  have  learned  that  the  area  of  any  rectangle  is  equal 
to  the  product  of  the  length  and  width. 

area  =  length  X  width  length  X  width  =  area 

Using  the  initial  letters  of  these  words,  this  principle  may 
be  expressed  in  the  following  formulas: 

a  =  Iw  or  Iw  =  a 

171.  Solving  Formulas.  To  solve  a  formula  completely 
is  to  find  the  value  of  each  general  number  in  terms  of  the 
others. 

Dividing  both  members  of  Iw  =  a,  first  by  I  and  then  by 
w,  we  obtain  the  two  new  formulas : 

w  =  T        and         1  =  — 
1  w 

This  formula  may  be  stated  in  words,  thus, 

Either  dimension  of  any  rectangle  is  equal  to  the  area  divided 

by  the  other  dimension. 

This   holds   only    when   the    area  and    the    given   dimension   are 

expressed  in  the  same  units  of  measure. 

Exercise  60 

1.  If  a  rectangular  lawn  48  feet  long  contains  1728  square 
feet,  what  is  its  width? 

1728 


126  ELEMENTARY   ALGEBRA 

2.  When  a  rectangle  18  feet  wide  contains  150  square 
yards,  what  is  its  length? 

9X150 
18 

3.  A  rectangle  of  land  64  rods  long  contains  18  acres. 

Find  its  width  in  rods. 

160X18 

w  = 

64 

4.  Express  in  general  numbers  two  rules  for  finding  the 
perimeter  of  any  rectangle. 

5.  Using  any  general  numbers,  write  three  formulas  for 
finding  the  area  of  any  triangle. 

6.  Solve  one  of  the  three  formulas  of  problem  5  and  give 
the  rule  which  each  of  the  derived  formulas  expresses. 

7.  If  a  triangle  whose  altitude  is  24  feet  contains  52 
square  yards,  how  long  is  its  base? 

8.  If  x  is  the  age  of  a  boy  now,  make  the  problem  of 
which  this  equation  is  the  statement: 

x+3  =  3(.T-7) 

9.  Using  any  general  numbers,  write  the  formula  for  find- 
ing the  volume  of  any  rectangular  prism. 

10.  Solve  the  formula  of  problem  9  and  give  the  principle 
which  each  of  the  three  derived  formulas  expresses. 

11.  Express  in  a  formula  the  relation  of  dividend,  divisor, 
quotient,  and  remainder,  in  division. 

12.  A  has  X  acres  of  land  and  B  3a:  acres.     Make  the 
problem  of  which  the  statement  is  3x  — 20  =  2(x+20). 

13.  Without  squaring  the  binomial,  give  the  difference 
between  (75+3)'  and  (75+3)(75-3). 

14.  Give  a  formula  for  finding  one  dimension  of  a  rectangle 
when  the  perimeter  and  the  other  dimension  are  given. 


FORMULAS  127 

15.  If  a  rectangle  64  feet  long  has  a  perimeter  of  226  feet, 
what  is  the  width? 

16.  Represent  the  number  of  cubic  yards  in  any  box- 
shaped  excavation  when  the  dimensions  are  given  in  feet. 

172.  The  formula  as  a  compact  shorthand  of  number  laws 
is  perhaps  the  most  practical  part  of  algebra.  The  following 
list  of  problems  will  give  practice  in  formulating  arithmetical, 
practical,  and  scientific  laws. 

Exercise  61  —  Stating  and  Formulating  Laws 

1.  Denoting  the  minuend,  subtrahend,  and  difference 
by  m,  s,  and  d,  respectively,  show  by  a  formula  the  relation 
of  these  numbers. 

2.  Add  s  to  both  sides  of  m  —  s  =  d  and  state  what  the 
resulting  formula  means. 

3.  Show  by  a  formula  the  relation  of  the  product,  p, 
multiplicand,  M,  and  multiplier,  m. 

4.  Divide  both  sides  of  p  =  M'm  by  m,  and  state  the 
meaning  of  the  resulting  formula. 

6.  State   as   a   formula:   "The   product   of   a   fraction, 

H 

- ,  by  a  whole  number,  a,  is  the  product  of  the  whole  number 
by  the  numerator,  divided  by  the  denominator." 

6.  Show  by  a  formula  the  principle  for  multiplying  a 

a  c 

fraction,  -,  by  a  fraction  -,  caUing  the  product  p. 
0  a 

7.  State  by  a  formula  the  relation  of  the  percentage,  p, 
the  rate,  r,  and  the  base,  h,  and  translate  the  formula  into 
words. 

8.  Divide  both  sides  of  p  =  br,  by  r,  and  tell  the  meaning 
of  the  resulting  formula. 


128  ELEMENTARY   ALGEBRA 

9.  State  and  give  meaning  of  the  formula  for  the  interest, 
i,  in  terms  of  the  principle,  p,  rate,  ?-,  and  time,  t  (in  years). 

10.  Divide  both  sides  of  i  =  prt  by  rt,  and  tell  what  the 
resulting  formula  means. 

11.  Divide  both  sides  of  i  =  prt  by  pt,  and  tell  what  the 
resulting  formula  means. 

12.  State  as  a  formula  the  law  for  subtracting  two  fractions. 

13.  State  as  a  formula  the  law  for  multiplying  two 
fractions. 

14.  Show  by  a  formula  the  law  of  area,  A,  of  a  square 
of  side,  s. 

15.  State  by  a  formula  the  volume,  V,  of  a  cube  whose 
edge  is  s. 

16.  State  by  a  formula  the  value,  J,  of  a  decimal  fraction 
having  t  units  in  tenths'  place  and  h  units  in  hundredths' 

place.  /       h 

Ans.     f  =  —  -\ . 

•      10     100 

17.  Solve  the  formula  in  the  answer  of  problem  16  for  t;  for/j. 

18.  State  as  a  formula  the  cost-law,  in  which  c  is  the  total 
cost,  n  the  number  of  articles,  and  p  the  price  of  each.  Solve 
the  formula  for  n;ior  p. 

19.  Calling  d  the  total  distance,  r  the  rate  of  movement, 
and  t  the  time,  state  the  distance-law  for  uniform  motion, 
as  a  formula. 

20.  Solve  the  formula  of  problem  19  for  r,  and  tell  the 
meaning  of  the  result.     Solve  for  t. 

21.  The  velocity,  v,  of  a  freely  falling  body  is  the  product 
of  the  gravity-constant,  g,  by  the  time,  t,  of  fall.  Formulate 
this  law.     Solve  it  for  g;  for  t. 

22.  Solve  the  formula,  A  =  2Tr{h+r)  for  7r;  for  zrr;  for  /i+r; 
for /i. 


FORMULAS  129 

23.  Formulate  the  principle:  ''The  reciprocal,  i?,  of  a 
number,  n,  shows  how  many  times  the  number  goes  into  1." 

24.  The  area,  A,  of  an  equilateral  triangle  of  side  a  is 

given  by  A=— v3-     State  this  formula  as  a  rule. 

25.  State  as  a  formula:  "The  value  of  a  fraction  is  not 
changed  by  multiplying  both  numerator  and  denominator 
by  the  same  number,  m. " 

26.  Write  as  a  formula:  "The  value  of  a  fraction  is  not 
changed  by  dividing  both  terms  by  the  same  number." 

27.  Write  as  a  formula:  "The  commission  equals  the 
product  of  the  rate  and  the  principal." 

28.  Formulate:  "The  area  of  a  parallelogram  equals  the 
product  of  the  base  and  altitude." 

29.  Give  the  meaning  of  the  formula : 

A  =  \/s{s  —  a){s  —  b)  (s  —  c), 
in  which  A  is  the  area  of  a  triangle,  a,  6,  and  c  the  lengths  of 
the  sides,  and  s  is  J  the  sum  of  the  sides. 

30.  Find  by  the  formula  of  problem  29  the  area  of  a  tri- 
angle whose  sides  are  6,  8,  and  10. 

31.  Give  the  meaning  of  the  formula  E=  ,  where  E 
is  the  energy  of  a  moving  mass,  Af ,  of  velocity,  Y . 

32.  Solve  the  formula  of  problem  31  for  M\   For  y^. 

33.  Solve -i-4  =  l  for  ^;  forP;  for^. 

ti      r      i^ 

34.  The  law  of  the  see-saw  board,  balanced  by  two  boys  is: 

di  and  c^  being  the  distances  from  the  support  of  the 
weights,  Wi  and  W2,  of  the  boys.  Translate  the  law  into 
words. 


130  ELEMENTARY   ALGEBRA 

FORMS  AND   TYPE-FORMS   OF   ALGEBRAIC   NUMBERS 

173.  Meaning  of  Type-Forms.  A  very  important  thing  to 
learn  in  algebra  is  the  meaning  and  use  of  forms  and  type- 
forms  of  algebraic  numbers.  By  the  form  of  a  number  is 
meant  how,  from  its  written  appearance,  it  looks  as  though  it 
were  made  up  out  of  simpler  numbers.  A  bit  of  valuable 
advice,  often  given,  but  seldom  appreciated  by  the  beginner, 
is  always  to  look  carefully  into  a  problem-  or  exercise  before  putting 
pencil  to  paper.  ''Look  before  you  leap"  is  a  good  motto 
for  the  young  algebraist.  Make  it  a  habit.  The  habit  is 
particularly  valuable  in  factoring.  The  amount  of  useless 
labor  it  will  save  you  will  compensate  many-fold  for  the 
effort.  The  way  to  start  the  practice  is  to  learn  what 
number-forms  mean  and  how  to  use  them.  This  is  not  an 
entirely  new  thing,  for  number-forms  are  used  early  in 
arithmetic. 

For  example,  when  you  learned  to  tell,  without  dividing, 
whether  5  is  a  factor  of  a  number,  by  noticing  whether 
it  ended  in  0  or  5,  you  were  using  the  form  of  the  number 
to  lighten  your  work. 

Likewise,  you  have  probably  learned  to  use  the  form  of 
a  number  to  decide,  without  dividing,  whether  the  number 
is  divisible  by  10,   100,  2,  4,  8,  etc. 

In  algebra,  an  acquaintance  with  number-forms  is  much 
more  useful  than  in  arithmetic. 

If  we  were  asked  to  indicate  the  sum  or  the  difference  of 
two  different  numbers  in  some  suggestive  form,  we  might 
write : 

(     )  +  (J     and     (     )-(     ), 

the  empty  curves  suggesting  that  any  numbers  whatsoever 
might  be  written  within  them.  But  while  these  forms  show 
sum  and  difference,  they  do  not  suggest  that  the  two  numbers 


FORMS   AND   TYPE-FORMS  131 

ill  question  are  to  be  different  numbers.  To  obviate  this 
objection  we  might  suggest  these  forms: 

(     )  +  [     ]     and     (     )-[    ], 

with  the  understanding  that  the  curved  and  the  square- 
cornered  symbols  are  to  suggest  that  different  numbers  are 
to  be  written  inside  the  differently-shaped  symbols. 

If  we  had  been  ingenious  enough  to  see  what  it  took 
mathematicians  hundreds  of  years  to  discover,  that  by 
simply  calling  one  number  x  and  the  other  y,  and  writing, 

x-\-y  and  x  —  y, 

we  have  everything  shown  easily  and  fully,  then  our  problem 
would  have  been  solved.  We  merely  remember  that  the 
different  letters  are  in  general  to  denote  different  numbers. 

174.  Examples  of  Type-Forms.  Any  other  letters,  as  a 
and  6,  might  as  well  have  been  used  as  x  and  y  in  the  last  sec- 
tion. But  X  and  y  are  easily  written,  and  serve  just  as  well 
as  any  other  letters,  so  algebraists  fall  into  the  habit  of 
using  them  more  than  others. 

We  say  then  that  x-\-y  and  x  —  y  are  respectively  the  forms 
for  the  sum  and  the  difference  of  any  two  different  numbers. 
Since  x-\-y  may  stand  for  (typify)  the  sum  of  any  two  num- 
bers^ it  may  be  called  a  type-form  for  the  sum.  Similarly, 
X  —  y  is  called  the  type-form  for  the  difference  of  two  numbers. 

The  type-form  for  the  sum  of  two  products  is  ax-\-hy; 
for  the  difference  of  two  products,  ax  — by. 

The  type-form  for  the  sum  of  two  products  having  one 
factor  common  to  both  products  is  ax-\-ay,  and  for  the  differ- 
ence of  such  products,  ax  — ay. 

The  type-form  for  the  sum  of  two  squares  is  x^+l/^,  and  for 
the  difference  of  two  squares,  x^  —  y'^.  Observe  that  x^+i/^ 
means  that  a  number  is  made  by  taking  two  different  num- 
bers, squaring  both,  and  adding  the  squares,  while  x^  —  y^ 
directs  us  to  form  a  number  by  choosing  two  different  num- 


132  ELEMENTARY   ALGEBRA 

bers,  squaring  both,  and  subtracting.  Clearly  then,  sucli 
short  forms  as  x^-\-i/  and  x'^  —  y^  are  very  compact  ways  of 
saying  a  great  deal. 

Such  a  number-form  as  x^-\-ax-\-b  is  the  type-form  for  num- 
bers to  be  built  up  by  choosing  a  number,  squaring  it,  adding 
the  product  of  it  and  some  second  number,  and  then  adding 
a  third  number.  As  x^-\-ax-\-h  has  three  terms,  it  is  a  tri- 
nomial. But  is  made  up  of  three  different  numbers,  x,  a, 
and  b.  Since  one  of  these  numbers,  x,  is  squared,  the  tri- 
nomial is  called  a  quadratic  (square-like)  trinomial. 

The  form,  x^-\-ax-\-h,  is  then  a  type-form  for  quadratic 
trinomials. 

175.  Tjrpe-Forms  Interpreted.  Since  x-{-y  stands  for  the 
sum  of  any  two  numbers,  if  we  multiply  it  by  itself  we  get 
the  square  of  the  sum  of  any  two  numbers.  Multiplying  x-\-y 
by  x-\-y  gives  us 

x^-\-2xy-hy^ 

1.  Hence  the  type-form  for  the  square  of  the  sum  of  two 
numbers  is  x^-i-2xy-\-y^.  As  a  type-form,  this  x^-\-2xy-\-y^ 
tells  us  much. 

1.  It  tells  us  that  the  square  of  the  sum  of  two  different 
numbers  is  a  trinomial. 

2.  It  tells  us  that  two  of  the  three  terms  of  the  trinomial 
are  made  by  squaring  the  numbers  to  be  added  separately. 

.  3.  It  tells  us  that  the  remaining  term  of  the  trinomial  is 
made  by  doubling  the  product  of  the  two  numbers  that  were 
added  to  give  the  original  sum. 

4.  It  tells  y^  that  a  shoi^t  way  of  getting  a  square  of  the 
sum  of  two  numbers  is  to  square  each  of  the  two  numbers, 
to  form  their  product  and  double  it,  and  then  to  add  the  three 
results. 


FORMS   AND  TYPE-FORMS  133 

Thus,  to  square  the  sum  10 -f- 5,  or  15,  calculate  10^,  5^,  and  2X5X 
10,  getting  100,  25,  and  100,  and  then  add  100,  25,  and  100,  getting 
225.     All  this  can  be  done  mentally. 

2.  Similarly,  x—y  multiplied  by  itself,  gives 

x'^  —  2xy-\-y'^ 
which  is  the  type-form  for  the  square  of  the  difference  of 
any  two  numbers. 

Thus,  38  =  40-2,  hence  38- =  (40 -2)2  =  40^-2x2x40  +  22  =  1600 
—  160+4=  1444.     Most  of  this  calculating  can  be  done  mentally. 

3.  Since  any  binomial  is  either  a  sum  or  a  difference,  x=i=y 
is  the  type-form  for  any  binomial. 

4.  The  type-form  for  the  square  of  any  binomial  is  then 

x'=t=2xy-\-i/ 

the  upper  or  lower  sign  being  used  according  as  the  binomial 
is  a  sum  or  a  difference. 

6.  The  type-form  for  the  difference  of  two  cubes  is  x^  —  y^. 

6.  The  type-form  for  the  sum  of  two  cubes  is  x^-{-y^. 

7.  The  type-form  for  the  difference  of  two  like  powers  is 
.T"  — ?/". 

8.  The  type-form  for  the  sum  of  two  like  powers  is  x"-f-?/". 

9.  The  type-form  for  the  product  of  the  sum  and  difference 
of  any  two  numbers  is  (x-\-y){x  —  y). 

10.  Give  in  words  the  meanings  of  the  type-forms  5  to  9. 


CHAPTER  XIV 
FACTORING 

176.  The  factors  of  a  number  are  the  numbers  whose 
product  is  that  number.  Factors  of  a  number  are  the 
makers  of  the  number,  by  multiplication. 

177.  From  the  law  of  the  algebraic  notation  and  the  mear  - 
ing  of  integral  exponents,  the  factors  of  a  monomial  are  the 
factors  of  the  coefficient  and  each  letter  as  many  times  as 
there  are  units  in  its  exponent.     Thus, 

6a^6-c  =  3  •  2  •  aaa  •  66  •  c  =  3  •  2aaahbc 

MONOMIAL   FACTORS 

Type-form:     ax-f-ay-faz 

178.  Polynomials  having  a  common  factor  in  every  term 
are  the  product  of  a  polynomial  and  a  monomial. 

By  definition  of  factors,  since  3a(2a  — 36)=6a2  — 9a6,  3a 
and  2a  — 3fc  are  the  factors  of  6a^  — 9a6. 

The  monomial  factor  is  the  greatest  common  factor  of  the 
coefficients  multiplied  by  the  lowest  power  of  all  the  common 
letters. 

Thus,    14a2  +  21rt  =  7a(2a+3)  and  mx^-\2x^  =  (Sx\{Zx^2). 

When  the  monomial  factor  is  one  term  of  the  polynomial, 
the  corresponding  term  in  the  polynomial  factor  is  1.  Thus, 
lox='  +  10x2-5a:  =  5a:(3:c2^2x- 1) 

179.  Rule. —  Divide  the  'polynomial  by  the  monomial  factor 
and  write  the  divisor  and  the  quotient  for  the  factors. 

Factors  may  always  be  checked  by  multiplying  them 
together  and  comparing  the  product  with  the  number  to  be 
factored. 

134 


FACTORING  135 

Exercise  82 
Factor  the  following  and  check  the  last  four: 

1.  da^-lOa"  2.  a^x^+a^x^  3.  ia^b-lOab^ 

4.  6x^+15x^  6.  xV-^y     .         6.  Uab^+7a^h 

7.  QaH^-\-Sa^x^-2ax^y  8.  'da^x^+2a^x^-4:a'xY 

9.  46V-8a6c2-4a26c3  10.  aa;y  -  Sa^x?/ -  a^x Y 

11.  6a2c2+9a26c2d-3ac2  12.  a^b^d-Sab^c'+a^bh^ 

COMMON   COMPOUND   FACTOR 

Type-form:     ax+ay+bx+by 

180.  The  terms  of  a  polynomial  may  sometimes  be  so 
grouped  as  to  show  a  common  compound  factor. 

Consider  ax-\-ay-\-bx-\-b2j 

The  first  and  second  terms  of  this  polynomial  contain  the 
common  factor  a,  and  the  third  and  fourth  terms  contain  the 
common  factor  6.  Grouping  the  terms  in  this  manner  and 
factoring  each  group,  we  have : 

a{x-\-y)-\-b{x-\-y) 

By  the  use  of  parentheses,  the  polynomial  is  thus  reduced 
to  two  terms,  which  are  similar  with  reference  to  the  com- 
pound factor,  x-\-y.  Combining  the  terms  according  to  the 
rule  for  addition  of  terms  partly  similar,  §72,  we  have: 

{a+b){x-\-y) 

The  first  term  is  not  always  grouped  with  the  second.  It 
may  be  grouped  with  the  third  term,  or  the  fourth. 

Factor  ax-\-bx-\-2a-\-2b,  grouping  the  first  with  the  third 
term,  and  the  second  term  with  the  fourth.     Thus, 

a{x-{-2)-hb{x-\-2) 


130  ELEMENTARY   ALGEBRA 

Exercise  63 

Write  the  factors  of  the  following  and  check : 

1.  ac  —  ad-\-cn  —  dn  2.  ax—  cy-\-cx  —  ay 

3.  ax-\-2x-\-ay-\-2y  4.  an-\-hn  —  ax  —  hx 

5.  a^-\-ahi-\-an'^-\-tf  6.  x^  —  'Mj  —  xy-\-Zx 

7.  a^—mn  —  an-\-am  8.  af'-\-aH-{-a'^x^-\-x^ 

181.  In  the  preceding  examples,  a  positive  monomial  factor 
is  taken  out  of  each  group.     Observe  the  following: 

ax-{-ay  —  bx—  by  =  (a  —  b)  {x-\-y) 

ax—ay—bx-\-  by  =  {a  —  b){x  —  y) 

Convince  yourself  that  the  equations  are  correct  by  multi- 
plying a  —  bhyx-\-y  and  a  —  bhyx  —  y. 

A  polynomial  cannot  be  factored  in  this  manner  unless  the 
compound  factor  is  the  same  in  each  group. 

To  get  the  same  compound  factor  in  each  group,  —6  is 
taken  out  of  the  second  group  in  each  of  the  two  examples 
above. 

Exercise  64 

Factor  the  following  polynomials  and  check: 

1.  an—b7i  —  ax-\rbx  2.  bx—by-\-y'^  —  xy 

3.  ax—by-\-ay  —  bx  4.  ab-{-xy  —  ay  —  bx 

6.  71^  — nx-\-ny  —  xy  6.  ax^  —  by-\-axy  —  bx 

7.  a}^-\-7n-x  —  ani-  —  aH  8.  abx  —  bc-\-ai  —  anx 

182.  In  some  cases  the  compound  factor  in  one  group  is 
like  the  remaining  terms  of  the  polynomial,  or  like  those 
terms  with  their  signs  changed.  In  such  examples  the 
monomial  factor  taken  out  of  one  group  is  +1  or  —1,  as, 
for  example, 

ax-ay-\-x-y={a-\-l){x-y) 
ax~ay-x-\-y^{a-l)(x-ij) 


FACTORING 
Exercise  65 

Factor  and  check  the  following : 

1.  ax+2x-\-a+2 

3.  3-c2+3c-c3 

5.  r'^+4-c2+4c 

7.  o  —  a'  —  r>o-{-a^ 

9.  rt^-ea—as^-e 

11.  l-7x3-x+7a;2 

13.  am-\-cn  —  an— cni  ■ 

15.  a(x-y)-b(y-x) 

17.  {a—c)a—(c—a)b 


137 


2.  ah  —  an-\-n  —  b 
4.  a^  —  a-\-ay  —  y 
6.  aa;H-6  — a  — 6a' 
8.  a6  — 6cH-a— c 
10.  ax-  — 6.T-  — a-f  6 
12.  rt-3a:2-|-3-aa:- 
14.  263+3-362-26 
16.  a3-12-2a4-6a2 
18.  a^  —  x*-\-a'^x  —  ax^ 

183.  Some  polynomials  may  be  separated  into  three  or 
more  groups  that  contain  a  common  compound  factor,  as  with 

ax  —  ay—hx-^hy-{-x  —  y-{a  —  h-{-l){x—ij). 

Exercise  66 
Factor  and  check  the  following: 

1;  ax  —  a  —  hx-\-h—nx-\-n 
2.  a;3-5x2-4a;+20  3.  a3-3a-f2a2-6 

4.  ax—hx—x  —  ay+by-\-y 
5.  4a6+c  — 6— 4ac  6.  a^  —  a—a%n-{-bn 

SQUARE   OF  THE  SUM   OF  TWO   NUMBERS 
Type-form:  a2+2ab+b2 

184.  Since  a  and  6  are  any  two  numbers,  (a-\-by  is  the 
square  of  the  sum  of  any  two  numbers.  The  square  of  a-f-6 
is  found  by  multiplication  to  be  a^-\-2ab-^b^,  or  the  square 
of  a,  plus  twice  the  product  of  a  and  6,  plus  the  square  of  6,  or 

(a+b)2  =  a2-f2ab4-b2 


138  ELEMENTARY   ALGEBRA 

185.  The  square  of  the  sum  of  two  numbers  is  the  square  of 
the  first  number,  plus  twice  the  product  of  the  first  and  second, 
plus  the  square  of  the  second. 


Exercise  67 
Give  the  results  of  the  following: 

1.  (6+c)2  2.  (x-hl)(x-hl)  3.  iax+h)~ 

4.  If  a  man  lives  8  years,  he  will  be  n  years  old.     How  old 
was  he  8  years  ago? 

5.  (a+cY  6.  (n+3)(n+3)  7.  {a+byY 

8.  What  will  represent  the  sum  of  3  consecutive  odd  num- 
bers of  which  s  is  the  smallest? 

9.  {b+xY  10.  Cr-f2)(x+2)  11.  {2a-\-by 

12.  A  man  was  x  years  old  a  years  ago.     If  he  lives,  how 
old  will  he  be  in  b  years? 

13.  (x+yY  14.  (n+4)(n+4)  16.  {x+SijY 


SQUARE    OF   THE   DIFFERENCE    OF   TWO    NUMBERS 

Type-form:   a--2ab+b2 

186.  Since  a  and  b  are  any  two  numbers,  {a  — by  is  the 
square  of  the  difference  of  any  two  numbers.  The  square  of 
a  —  b  is  found  by  multiplication  to  be  a^  —  2ab+¥,  or  the 
square  of  a,  minus  twice  the  product  of  a  and  b,  plus  the 
square  of  b,  or 

(a-b)2  =  a2-2ab+b2 

187.  The  square  of  the  difference  of  two  numbers  is  the 
square  of  the  first  number,  minus  twice  the  product  of  the  first 
and  second,  plus  the  square  of  the  second. 


FACTORING  139 
Exercise  68 
Give  the  results  of  the  following,  without  multiplying : 

1.  (b-cy                 2.  (n-l)(n-l)  3.  {ax- by 

4.  {a-cy                 6.  {x-S)(x-3)  6.  (a-btjy 

7.  {b-xy                 8.  (n-2){?i-2)  9.  (3x-4y 

10.  {b-yy               11.  (.x-4)(a;-4)  12.  {x-Syy 

13.  (a-a;)2               14.  (n-6)(?i-6)  16.  (4a-5)2 

188.  An  arithmetical  number  may  be  squared  mentally 
by  considering  it  to  be  the  sum  or  the  difference  of  two 
numbers.     Thus, 

462  =  (40+6)2  =  1600+480+36  =  21 16 
462  =  (50-4)2  =  2500-400+16  =  2116 

Exercise  69 

,  Express  the  squares  of  these  numbers,  first  as  the  sum, 
then  as  the  difference  of  two  numbers : 

1.  382  2.  472  3.  652  4.  542  6.  732 

6.  582  7,  642  8.  762  9,  352  ^q    952 

189.  A  trinomial  may  be  squared  by  grouping  two  terms 
to  make  a  binomial  of  it.     Thus, 

(a+6+c)2  =  a2+2a6+62+2c<a+6)+c2,  and 
(a+6+c)2  =  a2+2a(6+c)+62+26c+c2 

Exercise  70 

Give  the  following  squares  without  actually  multiplying : 
1.  (a+6-c)2  2.  (a-6+c)2  3.  (a^+c)2 

4.  (a-^+2)2  6.  (^+^-2)2  6.  {a+x^y 

190.  If  a  number  is  the  product  of  equal  factors,  one  of 
those  factors  is  called  a  root  of  the  number. 


140  ELEMENTARY   ALGEBRA 

The  square  root  of  a  number  is  one  of  the  two  equal 
factors  whose  product  is  the  number. 

Give  the  square  root  of  9;  25;  64;  144;  a^;  x';  4¥; 
16xy;    lOOa^^^c^;    (a+l)^;  (x-2)^ 

The  cube  root  of  a  number  is  one  of  the  three  equal 
factors  whose  product  is  the  number. 

Give  the  cube  root  of  8;  27;  64;  125;  a^;  ¥;  Sx^; 
21Qa^¥;    21a^¥&]    («+&)';    {x^yf. 

m 

TRINOMIAL    SQUARES 

Type-form:   x-=*=2xy+y- 

191.  A  trinomial  square  is  the  square  of  a  binomial.     Thus, 

^    {x^yY  =  x''-\-2xy+if 
(x-yy  =  x^-2xy+y- 

Two  terms  of  every  trinomial  square  are  the  squares  of  the 
two  terms  of  the  binomial. 

The  other  term  is  twice  the  product  of  the  square  roots  of 
the  two  squares  and  may  be  either  positive  or  negative. 

The  factors  of  a  trinomial  square  are  therefore  two  like 
binomials,  and  the  terms  of  each  factor  are  the  square  roots  of 
the  two  squares  in  the  trinomial.     Thus, 

4a2+962+12a6=(2a+36)(2a4-36) 

The  two  squares  are  4a2  and  9¥,  their  square  roots  are  2a  and  36, 
and  I2ab  is  twice  the  product  of  these  square  roots. 

The  method  of  factoring  a  trinomial  square  is  stated  as  a 
rule,  thus: 

192.  Rule. —  Find  the  square  roots  of  the  two  terms  that  are 
squares,  connect  these  roots  with  the  sign  of  the  other  term, 
and  write  the  binomial  twice  as  a  factor. 

The  two  factors  of  a  trinomial  square  being  equal,  it  is  evident  that 
each  factor  is  the  square  root  of  the  trinomial. 


FACTORING  141 

To  determine  whether  a  given  trinomial  is  a  square,  look 
first  to  see  if  it  contains  two  squares,  which  are  positive.  If 
two  of  the  terms  are  squares,  find  the  square  root  of 
each,  multiply  one  root  by  the  other  and  that  product  by  2. 
If  the  result  is  the  remaining  term  of  the  trinomial,  the  tri- 
nomial is  a  square.     For  example, 

a2-8a-[-16  =  (a-4)(a-4),  or  (4-a)(4-a) 

Each  of  the  first  two  factors,  a  —  \  and  a  — 4,  multiplied  by  —  1, 
gives  one  of  the  factors  of  (4 -a) (4 -a).  The  number  (a -4) (a -4), 
multiplied  by  (  — 1)(  — 1),  which  equals  +1,  gives  the  same  product  as 
(4— a)(4-a),  or  (a-4)(a-4)  =  (4-a)(4-a).  Consequently,  either 
pair  of  factors  is  correct. 

From  the  example  the  following  important  principle  may 
be  stated: 

193.  Principle. — The  signs  of  an  even  number  of  factors  may 
he  changed  without  changing  the  product. 


Exercise  71 

Determine  whether  the  following  are  trinomial  squares  and, 
if  so,  give  the  factors: 

1.  x2-4a:-f4  2.  ^a'^x^^Wx^X 

3.  a2-j-2a-f  1  4.  16-f-16.T^+4.T« 

6.  c*-M-2c2  6.  364-9m2+36m 

7.  4a2-f  H-4a  8.  ^a^+^x'-\-Wx^ 
9.   -6a+9a2-hl  10.  a^^-\W-Wh 

11.  a^+2a2x+x2  12.  Oa^-f  9c2-18ac 

13.  x^+?/-a;2|/  14.  -20r^+25x^+4x2 

16.  x^-{-y^-2xhf  16.  Qa^^  1662+ 24^2/^ 

-  17.  .T4+12r^-f  36.r2  18.  10()-20a6+a'-62 


142  ELEMENTARY  ALGEBRA 

19.  (a-\-xy-2ia-\-x)-\-l 
20.  IQx^+x^+Sx'  21.  9a462+30a26-f  25 

22.  (a+6)2+4(a+6)+4 
23.  4a*-h4a26-f  62  24.  2oa6c^+10a3c2-f  1 

26.   -6(x-2/)  +  (a;-i/)2+9. 
26.  64a;2+32x+4  27.  121+4a'^62-44a6 

PRODUCT   OF   THE   SUM   AND    DIFFERENCE    OF   TWO    NUMBERS 

Type-form:     (a+b)(a-b) 

194.  Letting  a  and  b  represent  any  two  numbers,  then 
{a-i-h){a  —  h)  represents  the  product  of  their  sum  and  differ- 
ence. The  product  is  found  by  multipUcation  to  be  a-  —  ¥, 
the  difference  of  their  squares,  i.e., 

(a+b)(a-b)=a2-b2 

195.  The  product  of  the  su7n  and  difference  of  two  numbers 
is  the  difference  of  their  squares. 

Exercise  72 

Give  the  following  products: 

1.  {a-{-c)(a-c)        2.  (x-l)(x+l)  3.  (2a-2)(2af2) 

4.  (c+6)(6-c)         5.  (n-2)(2H-n)  6.  (3x  +  l)(3x-l) 

7.  {a-\-x){a-x)       8.  (x-3)(3+x)  9.  (3-2a)(3+2a) 

10.  (6-x)(x+6)      ll.(4-a)(a+4)  12.  (2x+4)(2a:-4) 

1%.  The  product  of  two  arithmetical  numbers  may  be 
found  by  writing  them  as  the  sum  and  difference  of  the  same 
two  numbers.     Thus, 

26  X 14  =  (20+6)  (20  -  6)  =  400  -  36  =  364 


FACTORING  143 

Exercise  73 

Give  these  products  as  the  products  of  the  sum  and  the 
difference  of  two  numbers: 

1.  38X22  2.  47X33  3.  54X46 

4.  66X54  5.  72X68  6.  83X77 

197.  Two  trinomials  ma.y  sometimes  be  grouped  so  as  to 
represent  the  sum  and  difference  of  two  numbers.    Thus, 

{a-i-h-\- c){a+.b-c)  =  (^+6+  c)(^b-  c) 
{a-\-b-c){a-b^c)  =  {a-\-b^c)ia-b^c) 
Hence,     {a  +  b- c)(a-b+c)=a'-b''+2bc- c'' 

Exercise  74 
Give  the  following  products: 

1.  {a-{-x-\-y){a-\-x-y)  2.  {a-b-{-c){a-b- c) 

3.  {a+x-y)(a-x-\-y)  4.  {a-b+x)(a-\-b-x) 

5.  (x-\-y-2){x-y-\-2)  6.  (a+6+3)(a-fe-3) 

DIFFERENCE   OF   TWO    SQUARES 
Type-form:     a-  — b- 

198.  By  multiplying  the  sum  of  two  numbers  by  their 
difference,  the  difference  of  their  squares  is  obtained,  thus: 

(a:-h2)(a:-2)=x2-4 

Since  the  terms  of  the  product  are  squares  of  the  corres- 
ponding terms  of  the  factors,  the  terms  of  the  factors  are 
the  square  roots  of  the  terms  of  the  product,  or 

x''-l  =  {x-\-l){x-\) 

199.  Rule. —  Write  for  the  factors  the  sum  and  the  difference 
of  the  square  roots  of  the  terms  of  the  binomial. 


144  ELEMENTARY  ALGEBRA 

Exercise  75 
Factor  the  following: 

1.  a^-x-  2.  xr-9  3.  9x^-y- 

4.  rt«-6«  6.  l-a2  6.  4x^-y~ 

200.  When  both  terms  of  the  difference-factor  are 
squares,  that  factor  may  be  resolved  into  two  other  factors. 
Thus, 

Exercise  76 
Factor  the  following: 

1.  a^-x^  2.  x^-1  3.  a^-9x* 

4.  a*-¥  6.  l-n^  6.  9?/- 16 

7.  x^-if  8.  x^-9  9.  25-92/<5 

10.  a2_58  11.  4-x2  12.  4a;«-49 

13.  x^-i/  14.  a:8-l  15,  64-9x« 

16.  a^-a^  17.  l-rc^  18.  96^-81 

19.  9a2-4  20.  a2-4  21.  4x^-25 

22.  9-4x2  23.  .T»-16  24.  9a^-4x2 

26.  9a«-4  26.  x^-Sl  27.  4^4- feV 

201.  One  or  both  of  the  squares  in  this  type  of  example  may 
be  the  square  of  a  binomial.     Thus, 

(a-6)2-c2=(a-64-c)(a-6-c) 

a2-(6-c)2  =  (a+6-c)(a-6H-c) 

and,        (a—by—{x+yy=(a~b-\-x+y)(a-b—x-y) 


FACTORING  145 

Exercise  77 
Give  the  factors  of  the  following  and  check: 

1.  ix-\-yy-lQ  2.  {a+by-ix-i-yy 

3.  25-(a-6)2  4.  {a-by-{c-iy 

6.  {x+yY-SQ  6.  {a-\-xy-(y-2y- 

202.  The  square  of  a  binomial  in  examples  like  these  may 
also  be  expressed  without  a  symbol  of  aggregation.     Thus, 
a'-\-2ab+¥-  c^=(a+by-c^ 
a2-h^-2bc-(^  =  a'-{b+cy. 

When  the  second  square  is  placed  in  a  parenthesis,  preceded  by  a 
minus  sign,  the  signs  of  the  terms  must  be  changed. 

If  a  polynomial  is  the  difference  of  two  squares,  it  usually 
contains  four  or  six  terms.  The  term  that  has  a  numerical 
coefficient  generally  indicates  what  terms  must  be  taken  with 
it  to  make  a  trinomial  square. 

If  three  of  the  terms  as  they  stand  in  a  given  example, 
make  a  trinomial  square,  this  square  must  be  placed  first. 

If  the  monomial  square  has  a  minus  sign  before  it,  or  if  the 
remaining  terms  make  a  trinomial  square  when  placed  in 
parentheses  with  a  minu^  sign  before  them^  the  polynomial 
is  the  difference  of  two  squares. 

Exercise  78 

Factor  and  check  the  following: 
1.  a2-2a-62-f-l  2.  b^-{-c^-cP-2bc 

3.  a2-624-26-l+2acH-c2 
4.  x^-2y-y^-l  5.  x^-\-y''-2xy-9 

6.  x^-y^+2x-\-l+2yz- z^ 
7.  x^-y^-2y-l  8.  a^-¥-(^-2bc 

9.  a^-c^-\-¥-{-2ab-(P-2cd 
10.  {x—l)a  —  (l—x)  11.  ac^  —  acy-^cy  —  y^ 


146  ELEMENTARY   ALGEBRA 

Exercise  79 — Review 

Give  the  factors  of  the  following: 
1.  9a*x*-y^  2.  a^b^-c^cP  3.  lOOx^-i/^ 

4.  {x-\-yy+S{x-\-y)  +  lQ 

6.  Sla^+a*-lSa^  6.  Qm^H- 4^2 +12/^/1 

7.  {a+xy-\-25-10(a+x) 

8.  m-{a-\-xy  9.  {a-by-(c-2y 

10.  (6+c)2+16(6+c)-f64 

11.  64a:8+ 16x7 -fa:«  12.  49x2-42x2/2+91/^ 

13.  -12(x+2/)+4(x+2/)2+9 

14.  9a^-¥c*  16.  a266-xY  16.  a4-14464 

17.  ax  — a+6x  — 6  — 2cx+2c 

18.  a'-x^+2a+l  19.  2xi/+9-x2-i/2 

20.  X2/+x-32/2-3?/-2i/-2 

21.  a(&-c)-(c-6)  22.  x3-4x2-f2x-8 

23.  a2(6-c)2+8a(6-c)  +  16 

24.  a2-6x-x2-9  25.  a2+26c-62-c2 

26.  (x-2/)2-2ac(x-?/)+a2c2 

27.  (a-l)2-i/2  28.  (a-i-ny-ix-iy 

29.  9(a-6)2+12c(a-6)+4c2 

30.  25x*-81  31.  16a4~16x4  32.  9x'-Sh/ 

33.  a2+2x2/+2ac-x2-i/2+c2 

34.  x2-?/2-4x+4  36.  a2+x2-c2+2ax 

36.  a^+¥-x^-y^-2ab-\-2xy 

37. '49a2H-14acH-c2  38.  25rt^-20a26+462 


FACTORING  147 

PRODUCT   OF  TWO   BINOMIALS   WITH   A   COMMON  TERM 

Type-form:    (x-f-a)(x-|-b) 

203.  Multiplying  x-{-a  by  x-\-h  and  uniting  the  two  terms 
that  are  similar  with  reference  to  x,  we  find : 

(x+a)(x+b)=x2+(a+b)x+ab 

The  first  term  of  the  product  is  the  square  of  x,  the  second  term  is 
the  product  of  x  and  the  sum  of  a  and  b,  the  third  term  is  the  product  of 
a  and  b. 

204.  Principle. —  The  product  of  two  binomials  with  a  com- 
mon term  is  the  square  of  the  common  term,  plus  the  product 
of  the  common  term  and  the  sum  of  the  unlike  terms,  plus  the 
product  of  the  unlike  terms. 

Exercise  80 

Give  the  following  products  without  multiplying: 

1.  {x-\-2)ix-^l)  2.  {2a+by  3.  (w-6)(n-5) 

4.  {x-2){x+l)  6.  (x-4yy  6.  (a-|-4)(a-l) 

7.  {x-4){x-S)  8.  (a+36)2  9.  (^-h9)(2/+5) 

10.  (a+6)(a-5)  11.  (4a:-2)2  12.  (n-4)(n+3) 

13.  (x-{-5)(x-h4)  14.  (x+byY  16.  (x+7)(x-4) 

16.  (6-6)(6H-3)  17.  (3a-4)2  18.  (n-8)(n-2) 

19.  (.T-f5)(a--4)  20.  {a-\-4xy  21.  (x-7)(x-f  1) 

22.  (rt-9)(a-h8)  23.  {Sx-^y  24.  {n-\-S)\n  +  Q) 

25.  {x-^){x-2)  26.  (a+6^)2  27.   {x-\-9){x-4) 

28.  (a-5)(a-2)  29.  (4a-4)2  30.  (a-7)(rt-h6) 

205.  The  product  of  two  arithmetical  numbers  may  some- 
times be  conveniently  found  by  expressing  them  as  binomials 
with  a  commo7i  term.    Thus, 

46X36  =  (40+6)  (40 -4)  =  1600+80 -24 
57X42  =  (oO+7)(oO-8)  =  2500-50-56 


148  ELEMENTARY   ALGEBRA 

Exercise  81 

Give  the  following  products: 

1.  38X23  2.  (40+6)(40+4)  8.  57X47 

4.  64X62  6.  (80-3)(80-2)  6.  76X62 

SPECIAL   QUADRATIC   TRINOMIALS 

Type-form:     x--fax+b 

.  206.  The  product  of  any  two  binomials  with  a  common  term 
is  represented  by  the  following  trinomial : 

x2+ax-hb 
It  is  evident  that  the  factors  of  such  a  trinomial  are  the 
two  binomials  of  which  it  is  the  product. 

The  first  term  of  each  factor  is  the  square  root  of  x^,  i.e.,  x, 
the  second  terms  are  the  two  factors  of  b  whose  sum  is  a. 
Similarly,  .T2-h9x+18=  (a-+6)(a;+3) 

x^-9x-\-lS==ix-Q)ix-S) 
x'--\-Sx-lS=(x-^Q)(x-S) 
x^-3x-lS=(x-Q)(x-^S) 
The  factors  of  b  whose  sum  is  a  in  these  four  examples  are,  in  order: 
+6  and  +3         +6  and  -3 
-6  and  -3         -6  and  -f-3 
If  the  third  term  of  the  trinomial  is  positive,  the  second  terms  of  the 
factors  have  Zifce  signs;. if  the  third  term  is  negative,  the  second  terms  of 
the /actors  have  unlike  signs. 


Exercise  82 

Give  the  factors  of  the  following:. 

1.  «2_7a-f-l2  2.  w2-n-12 

4.  a2-7a-18  6.  n--{-n-^0 

7.  a2-8a  +  lo  8.  n'-^-^n-^b 

10.  a2-f9a+20  11.  /<2-fn- 56 

13.  a2-a- 132  14.  n--\-2n-^ 


3.  .t2-17xH-30 

6.  x2-f  14a;4-48 

9.  a:2-llx-12 

12.  x2-M3x+12 

16.  x'-\\x^-m 


FACTORING  149 

16.  a2+6a-72  17.  n^-n-72  18.  a;2-hl7a;-18 

19.  a2+4a-21  20.  n^-Sn-9  21.  x^+lSa^+SG 

22.  a2+3a-40  23.  w2-|-n-42  24.  x2+15a;-16 

26.  a--\-4a-45  26.  ^|2-6^,_60  27.  x^-7x-ZZ 

28.  a2-.3a-70  29.  ^2  +  10/1-24  30.  a:2-12a:-45 

31.  a^-lla-eO  32.  n^-nn+m  33.  x'"-\-l5x-S4: 

THE   GENERAL   QUADRATIC    TRINOMIAL 

Type-form:  ax'+bx+c 

207.  Some  trinomials  of  the  form  ax^+bx+c*  are  the 
product  of  two  binofnials.     Thus, 

3x+     2 
2a;  +     3 
3-2x2-f2-2a: 

3-3a;+2'3 

Qx'+lSx-\-Q 

The  first  term  of  the  product  is  the  product  of  the  first  terms  of  the 
binomials;  the  last  term  is  the  pi'oduct  of  the  last  terms.  Hence,  the 
product  of  the  first  and  last  terms  of  the  trinomial  is  the  product  of  the 
four  terms  of  the  two  binomials.  The  two  terms,  whose  sum  is  13a;, 
also  contain  the  same  factors. 

We  conclude  that  we  may  resolve  such  a  trinomial  into  two  binomial 
factors,  provided  we  can  separate  the  product  of  the  first  and  last  terms 
into  two  factors  whose  sum  is  the  middle  term. 

For  example,  consider:       9x-+43j-  — 10 

The  product  of  the  first  and  last  terms  is  —90x^,  and  the  factors  of 
—  90x2,  whose  sum  is  43aj,  are  45a;  and  —2x.     Now  factor  thus: 

9x^+43x-m  =  9x^-{-A5x-2x-10 

=  {9x-2){x-\-5) 

If  the  last  term  of  the  given  trinomial  is  positive,  the  second  terms  of 
the  factors  have  like  signs;  if  negative,  they  have  unlike  signs. 

*Trinomials  of  the  form  ax--\-hx-\-c  are  called  general  quadratic 
trinomials. 


150  ELEMENTARY  ALGEBRA 

Exercise  83 

Give  the  factors  of  the  following: 

1.  3a2+a-2 

3.  4:X^—x  —  5 

5.  6a2+a-2 

7.  7x2+x-6 

9.  Sa^-a-9 
11.  6x2+x-5 
13.  2a2-5a+3 


2.  5x2-17x4-14 

4.  7a2-17a-12 

6.  8x2-45x-18 

8.  9a2+32a-16 

10.  4x2-8x2/+3t/2 

12.  6a2-5a6-662 

14.  8x2+6x2/ -92/2 


Exercise  84 

Factor  the  following: 

1.  a2-12a-28 

3.  l+6x-72x2 

5.  a2+19a+84 

7.  6x2+31x+35 

9.  x4-llx2-42 
11.  a4-14a2+45 
13.  x^- 21x2+80 

16.  3a2-13a-30 

17.  a2+16c2-8ac 
19.  l-8x+12x2 
21.  a2-9ac+8c2 
23.  9a2-8ax-20x2 

26.  x8+ 11x^-60 

27.  a6+15a3+56 


—  Review  of  Factoring 


2.  3x2-13x+12 
4.  2a2+5ax-3x2 
6.  81x«+ 16+72x3 
8.  49aH98a2+49 
10.  12a-12a2-9a3 
12.  18a2+3a6-45fe2 
14.  12162+886+16 
16.  8x2+49x2/-49?/ 
18.  l-a2-62+2a6 
20.  16I/2+ 162/^+322/2 
22.  12a2+31ax+9x2 
24.  36x8+25x2-60x^ 
26.  6x2+23x2/+2l2/2 
28.  49x2+70x2/+252/2 


FACTORING  151 

29.  15x^-\-4xij-4:y^  30.  S6n^-\-25x^-(y0nx 

31.  ¥-c^-2b^+2c(P-hl-d' 
32.  8ac-4a2-4c2+4  33.  b^-a^-c^-2ac 

34.  a2+4c2-9.T2+6x-l-4ac 
36.  4{a-x)-2{x-a)  36.  6^+762-36-21 

37.  a.T  — 6?/+z— 6a:-f-i/H-a2-)-a;+a?/— 62 
38.  a}h-ac'--ahm-\-chn  39.  2x-^-4x-3x2+6 

40.  ax—hy—z—hx-\-y  —  az-{-x-\-ay-\-hz 
41.  (a2-62-a:2)2-462x2  42.  l-a;2-^2_^2xi/ 

43.  ab-i-bx  —  by—ac—cx-\-cy-{-az-j-xz  —  yz 
44.   (a:2-?/+22)2_93.2^2  45^  16-4.t2+12cx-9c2 

46.  bm-\-bn  —  bp  —  cm—cn-\-cp-\-m-\-n—p 
47.   (m-p'^-xy-n'z-  48.  p^- 8^5-^2^ +3^2 

INCOMPLETE   TRINOMIAL   SQUARES 

Type-form:  x^+x^+Y^ 
208.  Some  trinomials  and  binomials  which  may  be  made 
trinomial  squares  by  the  addition  of  a  square  to  them  may  be 
resolved  into  two  trinomial  factors.     For  example,  consider: 

9a^+2a262+64 

This  trinomial  would  be  the  square  of  4a262,  if  the  coeffi- 
cient of  the  second  term  were  6.     Proceed  thus: 

9a4+2a262+6^ 

4a262        -4^252 

9a4+6a262+6^-4a262=(3a2+62+2a6)(3a2+62-2a6) 

Adding  40^62 _ 4^252^  which  equals  zero,  to  9a*-{-2a%^-\-b*,  the  value 
is  not  changed,  and  we  then  have  the  difference  of  two  squares. 


152  ELEMENTARY  ALGEBRA 

209.  When  the  second  term  of  the  trinomial  is  negative, 
two  different  squares  may  in  some  eases  be  added,  thus: 

a'b^        -a%^  9a^b^        -Qa^b^ 

4a4- 4^262+ 64-a262  4a''-\-4a^b^-{-¥-9a''b^  i 

The  factors  of  these  two  results  are : 

(2a2-62+a6)(2a2-62-a6) 

(2a2-f-62+3a6)(2a2+62-3a6) 

This  would  seem  to  indicate  that  the  expression  has  tioo  sets 
of  prime  factors,  but  this  is  impossible. 

We  find  that  each  of  these  factors  may  be  factored  by  the  preceding 
case,  §  207,  giving  the  following  factors: 

{2a-b)(a-\-b){2a+b){a-b) 

{2a+b)(a-\-b){2a-b){a-b) 

These  factors,  though  arranged  differently,  are  alike,  and  we  conclude 
that  when  two  squares  can  be  added  to  the  expression,  it  can  be  resolved 
into  four  biiiomial  factors,  and  it  is  immaterial  which  of  these  two  squares 
is  added  to  the  expression. 

When  a  binomial  can  be  factored  by  this  method,  it  can  generally 
be  resolved  into  four  binomial  factors. 

Exercise  85 
Factor: 

1.  x'-^i  2.  64a4-f-l  3.  x^-h4i/ 

4.  4x^-17x2+16  6.  36a^+24a2a;2+25a;^ 

6.  9a^-34a2+2o  7.  mx^-72xY+'^^i/ 

8.  x*-10xY+9y^  9.  Sla''-\-2Qa'b'"+25b' 

10.  a*~19a^b^-\-25b*  11.  SQx^-SSxY+'^^V* 

12.  9a*  -  8^262 -f  166*  13.  64x*-f  76x2|/2+492/* 

14.  Slx^^40xh/+4i/  16.  16a*-76a2.T2+25x* 


FACTORING  153 

DIFFERENCE    OF   THE   SAME    ODD   POWERS 

Type-form:  x^  — y^ 

210.  The  difference  of  the  same  odd  powers  of  two  num- 
bers is  the  product  of  a  binomial  and  a  polynomial. 

The  following  equations  may  be  verified  by  multiplication : 

x^  -  if  =(x-y)  (x- + xy + y^) 
x^  —  y^=(x~y  —  y^={x~  —  y){x*-^x^y-{-y^) 
x^  -  if  =  (x'-y  -  (?/)3  =  (.t2  -  y^)  (.t4+xV+2/®) 
3.5  _  y'.  =  (x-  y)  {x^  -\-x^y +xy  -\-xif + ?/) 

It  follows  that  such  binomials  may  be  resolved  into  two 
factors,  a  binomial  and  a  'polynomial. 

The  binomial  factor  is  the  difference  of  the  same  odd  roots 
of  the  two  terms  of  the  binomial. 

All  the  terms  of  the  polynomial  factor  are  positive. 

Exponents  of  x  and  y  in  the  polynomial  factor  decrease  and 
increase  by  the  exponents  in  the  binomial  factor. 

When  one  term  of  a  binomial  is  an  arithmetical  number,  it  may  some- 
times be  expressed  as  the  same  power  as  the  other  term. 

Exercise  86 
Give  the  factors  of  the  following: 

1.  a^-b^                    2.  .^3-27  3.  125-64x3 

4.  r^-1                      5.  a3_64  6.  216-27a6 

7.  a'-b'                     8.  n«-27  9.  1000 -Sx^ 

10.  1-x^                    11.  a»-64  12.  512-64x3 

13.  a'-b^                   14.  x»-27  16.  27-343x6 

16.  x3-8                    17.  a^-S2  18.  729-64a» 

19.  a^-x«                  20.  8x3-27  21.  27a^-QW 


154  ELEMENTARY  ALGEBRA 

SUM    OF   THE   SAME   ODD    POWERS 

Type-form:  x^-fy' 

211.  The  sum  of  the  same  odd  powers  of  two  iiumbers  is 
the  product  of  a  binomial  and  a  polynoniiaL 

The  following  products  may  be  verified  by  multiplication: 

x^-\-y^={x-\-y){x--xy-{-y-) 

x'-\-y'  =  x'+  {y'y  =  {x+f~){x'--xy'+y') 

a--^ +?/  =  (x-i-y)  (x^  -  x^y  -\-xh/  -  xy^-\-y^) 

The  hinomial  factor  is  the  sum  of  the  same  odd  roots  of  the 
two  terms  of  the  hinomial. 

The  terms  of  the  -polynomial  factor  are  alternately  positive 
and  negative. 

Exponents  in  the  polynomial  factor  decrease  and  increase 
by  the  exponents  in  the  binomial  factor. 

Exercise  87 

Give  the  factors  of  the  following: 

1.  a^+S  '          2.  n^+64  3.  512+640^ 

4.  a'+b^  5.  rc9+27  6.  27+343x« 

7.  l-\-¥  8.  a^+32  9.  729-f64x9 

10.  x^-\-y^  11.  8a:«+27  12.  27x''-\-Q4y^ 

13.  84-63  14.  a3+125  16.  Sa^+I25b^ 

16.  a^-x^  17.  x^+2m  18.  729x^-\-Sy^ 

19.  x^+l  20.  a;«-343  21.  lOOOa^+ft^ 

22.  a7_|-57  23.  a3_5i2  24.  Sx^-\-MSy^ 

26.  x'^-l  26.  a;^-f243  27.  Q4a^-\-27¥ 

212.  Some  binomials,  especially  those  that  are  the  differ- 
ence of  the  same  powers,  have  more  than  one  binomial  factor. 

The  binomial,  a^  —  x'^,  has  5  hinomial  divisors.  Show 
what  they  are  and  why  they  are  divisors  of  a^—x^. 


FACTORING  155 

213.  Summary  of  Factoring. 

I.  First  take  out  all  monomial  factors,  and  retain  their 
product  as  one  factor  of  the  given  expression. 

After  the  monomial  factors  are  removed,  next  notice  the 
number  of  terms  in  the  remaining  factor. 

II.  Binomials  are  factored  as : 

(a)  The  difference  of  two  squares,  thus, 

a2-b2=(a+b)(a-b) 

(6)  The  difference  of  the  same  odd  powers, 
a3-b3=  (a-b)(a2+ab+b2) 
a»-b^=  (a-b)(a^+a3b+a2b2+ab3+b^),  etc. 

(c)  The  sum  of  the  same  odd  powers, 

a3+b3=  (a+bX(a2-ab-hb2) 

as+b^=  (a4-b)(a4-a3b-ha2b2-ab-^-|-b^),  etc. 

III.  Trinomials  are  factored  as : 

(a)  A  trinomial  square,  thus, 

a2±2ab+b2=(a±b)(a:±=b) 

(6)  A  quadratic  trinomial, 

x^-f-ax+b,  or  ax^-j-bx+c,  by  inspection  and  trial, 

(c)  A  form  reduced  to  Ila,  thus, 

X^  +  Xy  +  y^=  (x2-4-y2)2_  (xy)2 

IV.  Polynomials  of  four  or  more  terms  are  factored : 

(a)  By  grouping  terms,  thus, 
ax+bx+ay+by=  (a+b)  x+(a+b)  y=  (a+b)(x+y) 

(6)  By  Ila,  thus, 

a'^+b2+2ab-c2=  {&+hy-c^=  (a+b-c)(a+b+c) 


156  ELEMENTARY  ALGEBRA 

REVIEW 

214.  Observe  the  following  rules  and  factor  the  exercises 
below: 

7.  //  the  expression  contains  a  monomial  factor,  that  factor 
should  in  every  case  be  removed  first. 

II.  All  binomials  should  first  be  factored  as  the  difference 
of  two  squares,  if  this  is  possible. 

III.  Do  not  write  a  compound  factor  as  one  of  the  factors 
of  an  expression  until  you  are  sure  it  is  prime. 

Exercise  88  —  General  Review  of  Factoring 
1.  a*+15a2+44  2.  8a2+37a6-1562 

3.  x*—x  4.  a^  — 81  5.  x^-x^y+xy 

6.  x^+lOx^-n  7.  50a2-35a6-462 

8.  x^-\-2xy-\-y^  —  4:X  —  4:y 
9.  9a2-18a6+862  10.  a;«-4x^-32 

11.  ax-{-cy+x  —  ay—  cx  —  y 
12.  a^-7n^-\-an-mn  13.  20a;2-|-8x-9 

14.  5a2+8a6-h362+5a+36 
16.  25a2-9(36-2c)2  16.  IM'-ax-^x^ 

17.  x'-j-y^-^-2z^-2xy^-l 
18.  x^-\-y^  19.  5a;¥-5xY  20.  r^-216 

21.  (a+x)2-l-2x(a+a;-l) 
22.  4x^-61xy-|-81i/«  23.  a^-a'^-a^+a^ 

24.  a2-62_c2+26c+a-6H-c 
26.  (a2-|-a;2-2/2)2-4aV  26.  l  +  19a:'-20a'^ 

27.  a2-f.4c+l  +  62-2a6+4c2 
28.  3(a-6)-(6-a)  29.  Sc^- c'--dc-\-l 

30.  Sx^-^xy-\-Sy^+Sxz-dyz 


FACTORING  157 

31.  a3-63-3a26-3a62  32.  9x^+l0xY+y^ 

33.  xY-x^-9y^-\-9  34.  a^-l-x^-2x 

36.  x^-z--\-2x-y'^+l+2yz 
36.  5ac^-dac  37.  x^'^+^f'^  38.  27a;3+l 

39.  (a;2-3a;)2-6(x2-3a;)+8 
40.  a8+19a46+8462  41.  8c2-31c+4 

42.  (a2-2a)+6(a2-2a)4-9 
43.  a(a:-2)-2(2-a:)  44.  c-y+cx-xy 

46.  c2-2/2-d2_2rf?/-2cx+x- 
46.  'SQx^-lQxY+y*  47.  6a'- -11a -72 

48.  2xy+c^+d''-x'+2cd-i/ 
49.  x^^- 2* +2x2?/ +1/2  60.  x^+xiz-o;-?/ 

61.   c2-n2+x2+26n-62+2cx 
62.  a;^-10a;2/-24i/8  63.  12x2+37x-10 

64.  (x2+5x)2-12(x2+5a:)+36 
66.  a8-16  66.  x^o-i/io  67.  3a^-3a 

68.  x^  —  y^—n^  —  2ny -f  /Ti^  _  2fnx 
69.  25a:4-94x2i/-f81?/4  60.  2c-c2+62_i 

61.  2ax-^'Sbx-\-cx  —  2ay  —  Shy—cy 
62.  a"*  — a^c^+a^x^- c^x^  63.  xy  —  iix-\-y—n 

64.  x^-\-ax—bx  —  2xy  —  ay-\-by-\-y^ 
66.  (a-6)2+(a-6)-2  66.  a2-c2-h4-4a 

67.  ?/i2_^4_4^^^_9^2_|_552^_^4c2 

68.  a{a-c)-{c-a)  69.  20x2-21.r-54 

70.  2aa;  — 36x+4cx  — 2a2/+36^  — 4c?/ 
71.  (a+6)2-2c(a  +  6)-f  c2  72.  a-d-a^+^a" 


Second  Half- Year 


CHAPTER  XV 

EQUATIONS.    EXERCISES  FOR  REVIEW  AND 
PRACTICE 

SOLUTION   OF   EQUATIONS   BY  FACTORING 

215.  Many  equations  containing  the  square  of  the  un- 
known number  may  be  solved  by  means  of  factoring.     Thus, 

7x2-8  =  4x2+19 

If  we  transpose  all  terms  to  the  first  member,  unite  the 
terms  containing  x^,  and  divide  both  members  of  the  equation 
by  the  coefficient  of  x^,  we  have  the  following : 

x2-9  =  0 

Factoring  the  first  member,  the  result  shows  the  indicated 
product  of  two  factors  equal  to  0. 

(x+3)(x-3)=0 

A  product  is  0,  if  one  of  its  factors  is  0.  Since  this 
product  (x+3)(x  — 3)  is  0,  at  least  one  factor  of  it  is  0. 

If  x+3  =  0,  then  x=-3;  and  if  x-3  =  0,  then  .t  =  3. 
Since  both  numbers  satisfy  the  equation,  x=+3  and  —3. 
Both  are  roots. 

It  is  important  to  notice  here  that  equations  containing 
the  square  of  the  unknown  number  have  two  roots. 

216.  The  statement  in  itaUcs  can  be  made  clearer  by  a 
little  graphing. 

158 


SOLUTION   OF  EQUATIONS   BY   FACTORIN(J         159 


I.  We  begin  by  showing  the  graphical  solution  of 

x'-i  =  0  (1) 

We  first  calculate  and  make  the  graph  of  the  first  member 


4. 


Thus  calculate: 

=  1         (x       =2       (x 
=  -3     \x'--i=Q      \x2-4 

fa:         =-2 
\x'-- 


{'■ 


=  3 
=  5 


(x         =0  ix 

\x'-4=-4:       \x2 


h 


-3 


X' 


\     -    7 

5     "W^ 

V         u 

.^o¥ 

\  "t. 

— ^4^ — 

lop 

0  \x2-4  =  +5 

Graphing  the  a;-values  horizontally  and  the  corresponding 
a:-— 4-values  vertically,  and  connecting  the  points  with  a 
smooth  curve,  we  obtain  the  curve  of 
the  figure. 

Equation  (1)  really  asks,  ''What  value 
or  values  of  x  make  x^— 4  =  0?"  Since 
the  x^— 4-values  are  the  vertical  distances 
from  the  horizontal  to  the  curve,  this 
question  amounts  to  asking,  ''What  are 
the  x-values  where  the  curve  crosses  tha 
X-axis?"  The  answer  is  seen  from  the 
figure  to  be  x=  +2  and  x=  —2. 

Both  +2  and  -2  substituted  for  x  in  x^ -4  =  0  satisfy  it. 
For  this  equation  then  there  are  two  values  of  x,  because 
the  curve  crosses  the  horizontal  in  two  points. 

Observe  that  we  graph /(x)  =^2— 4  and  obtain  the  parabola. 
The  solutions  of  /(x)=0  are  the  x-values  of  the  crossing- 
points  of  the  parabola  over  the  horizontal  axis. 

II.  Let  us  now  solve  graphically  the  equation, 
a:2+3a:  =  0  (2) 

Graph  the  first  member,  x^-f  3x,  first  calculating: 


Y' 

Scale 
1^1   horizontal  space 
2^1   vertical   space 

Graph  of  x^-i 
A  Parabola 


x  =  l, 


3, 

18, 


-1, 

-2, 


-2, 
-2. 


-3, 
0, 


and  —4 

and  +4 


160 


ELEMENTARY   ALGEBRA 


X' 


S 


ffl 


w 


Graphing  and  connecting  the  points, 
we  get  the  graph  of  x^+Sx  as  in  the 
figure.  Clearly  the  x-values  of  the 
crossing-points,  i.e.,  the  points  where 
a;2+3x  is  equal  to  0,  are  x  =  0  and 
x=— 3.  These  values  both  satisfy 
a;2H-3x  =  0,  and  we  again  have  two 
values  because  there  are  two  crossing- 
points. 
Here  we  graph  f{x)=x^-\-'i^x  obtaining  a  parabola,  whose 

crovssing-points   over    the   horizontal    give   the   x- distances 

that  are  the  solutions  of  f{x)  =  0. 

III.  The   graphical  solution  of  the  more  general  form 


Y' 

StaJe 
1    =    I    horizontal  space 
2=1    Terlical   space 

Graph  of  x'^-\-Zx 
A  Parabola 


a:2-6a:-f8  =  0 
is  obtained  by  first  graphing  x-  — 6x+8. 
Calculating  the  values: 


(3) 


x  =  h 

2 

3, 

4, 

5, 

6, 

0, 

-1 

2-6a:4-8  =  3, 

0, 

-1, 

0, 

+3, 

+8, 

+8, 

-f-15 

The  curve  is  shown  in  the  figure  and 
there    are    again  two  crossing-points, 

x=+2  and  x  =  -\-A,  and  these  satisfy 

a:2-6x+8  =  0. 

For  any  equation  containing  the 
square  of  the  unknown,  the  graph  of 
the  first  member  would  be  such  a 
curve  as  we  have  found  above. 

Hence,  equations  containing  the  square 
of  the  unknown  have ,  in  general,  two  roots. 

The  parabola  is  also  the  graph  of  f(x)=x'^  —  Qx-\-S,  and 
its  crossing-points  over  the  horizontal  give  the  solutions 
of/(x)  =  0. 


y         Scale 
I     =    1    hoiizontal  space 
2=1    vertical  space 

Graph  of  x^-Qx-{-S 
A  Parabola 


SOLUTION   OF  EQUATIONS  BY   FACTORING         161 

In  conclusion,  whatever  the  function,  f{x)j  that  forms 
the  first  member,  we  graph /(x),  and  the  crossing-points  of  the 
graph  over  the  horizontal  give  the  solutions  of  the  equation 

/(x)  =  0 

Transposing  all  terms  to  the  first  member,  solve  graphi- 
cally some  of  the  exercises  from  1-10  below. 

Exercise  89 

Solve  the  following  equations  by  factoring  as  the  equation 
of  §  215  was  solved: 

1.  a:2-13  =  19-x2  2.  5y^-50  =  Sy^-\-22 

3.  5x2-21  =  3 -x2  •      4.  68-h3?/  =  4?/-76 

6.  x24-31  =  5.r2-5  6.  Gi/^ -  98  =  3?/ - 95 

7.  2x2-|-5  =  x24-30  8.  48H-6?/2  =  7y2_52 
9.  a;2-|-90  =  3x2-8  10.  9?/2-48  =  4t/2-43 

11.  The  square  of  a  certain  number  increased  by  the  square 
of  3  times  the  number  is  640.     Find  the  number. 

12.  The  smaller  of  two  numbers  is  ^  of  the  larger,  and  the 
difference  of  their  squares  is  648.     Find  the  numbers. 

13.  The  square  of  a  number  increased  by  32  equals  twice 
the  square  increased  by  7.     Find  the  number. 

14.  Three  times  the  square  of  a  number  equals  4  times 
the  square   diminished  by    142.     Find  the   number. 

16.  One  numbei-  is  double  another  and  the  sum  of  the 
squares  of  the  two  numbers  is  125.     Find  the  numbers. 

16.  One  number  is  n  times  another,  and  the  sum  of  their 
squares  is  IG-flGn^.     Find  the  numbers. 

17.  Four  times  the  square  of  a  number,  diminished  by  76, 
equals  68  increased  by  3  times  the  square  of  the  number. 
Find  the  number. 


162  ELEMENTARY   ALGEBRA 

Exercise  90  —  Questions  and  Oral  Work 

Answer  the  questions  in  number  symbols  and  perform 
the  operations  indicated  in  the  even  numbered  exercises. 

1.  What  may  represent  the  area  of  any  rectangle  the 
length  of  which  is  twice  the  width? 

2.  {x-}-l){x-l)  {2x+Syy  (a,+2)(a-l) 

3.  How  many  days  will  it  take  a  man  to  build  m  yards  of 
wall,  if  he  builds  n  feet  a  day? 

4.  {a-n)(a-n)  {3a-Uy  (x-f-3)(a:+l) 

5.  If  a  man  lives  x  years,  he  will  be  y  years  old.     How  old 
was  he  c  years  ago? 

6.  (x+3)(x-3)  {4x-\-oijy  (a-f5)(a-5) 

7.  Express  4  times  the  product  of  a  square  and  x  cube, 
increased  by  n  times  the  number  2x  —  Sy. 

8.  ia-y){a+y)  {2a-ahY  {x-b){x-b) 

9.  If  2x  — 5  represents  an  odd  number,  what  will  represent 
the  next  smaller  odd  number? 

10.  (a+7)(aH-3)  {xy^oxf  .(4+a^)(a:-4) 

11.  Express  7  times  the  third  power  of  x,  diminished  by 
3  times  the  sum  of  2a  and  56. 

12.  (x+3)(x-2)  (3a-a6)2  (a-5)(a-f4) 

13.  A  boy  has  x  silver  dollars,  y  dimes,  and  z  5-cent  pieces, 
$5.80  in  all.     What  expression  equals  580? 

14.  (x-2)(a:+2)  {xy+^yf  (x-7)(x+3) 

15.  If  a  park  is  I  rods  long  and  w  rods  wide,  how  many 
times  must  a  man  walk  around  it  to  travel  n  miles? 

16.  {a-n){a-{-n)  {^a-ahf  (n-9)(n-5) 

17.  If  a  square  is  formed  by  adding  3  feet  on  all  sides  of  a 
smaller  square,  how  many  square  feet  are  added? 


SOLUTION   OF  EQUATIONS   BY   FACTORING         163 

Exercise  91 

Simplify  the  first  six  of  the  following  equations  and  solve 
the  rest  by  factoring: 

I.  3a;2-15a2  =  2x2-lla2  2.  Sy^-{-n¥=4y^-12b^ 
3.  2x2+17a2  =  5a;2-10a2  4.  5y^-40ri'  =  Sif+S27f- 
6.  7a^2_  43^2  =  32.2.^16^2                    g    6?/+2462  =  8?/+22/r 

Some  equations  that  contain  both  the  first  and  the  second 
powers  of  the  unknown  number  may  be  solved  by  factoring. 

a;2-ha;-12  =  0  4x^-{-l9x-5  =  0 

(x+4)(aJ-3)=0  {x+5)i4x-l)=0 

0:=— 4  and  3  a:=— oandj 

7.  The  square  of  a  certain  number  diminished  by  3  times 
the  number  is  130.     Find  the  number. 

8.  9a:2-6a:-12  =  3x2  9.  32/2-3?/-25  =  2t/2-15 

10.  The  product  of  3  times  and  5  times  a  certain  number 
is  735.     Find  the  number. 

II.  8x2+4x-80  =  4a:2  12.  5n2+14n-6  =  3n2-18 

13.  If  to  the  square  of  a  number  the  number  itself  is  added, 
the  sum  is  240.     Find  the  number. 

14.  5a:2+21a;+36  =  2a:2  16.  7y^-{-6y-45  =  2y^-15 

16.  The  sum  of  the  squares  of  two  consecutive  even  num- 
bers is  580.    Find  the  numbers. 

17.  Sx^+27x-{-42  =  5x'^  18.  82/2-32i/+8  =  4?/-52 

19.  The  quotient  of  one  number  divided  by  another  is  4, 
and  their  product  is  256.     Find  the  numbers. 

20.  The  length  of  one  square  field  is  twice  that  of  another, 
and  both  together  contain  1280  square  rods.  What  is  the 
length  of  each  side  of  the  smaller  square? 


164  ELEMENTARY   ALGEBRA 

EXERCISES  FOR  REVIEW  AND  PRACTICE 
Exercise  92  —  Oral  Practice 

Answer  in  number  symbols  and  perform  indicated  oper- 
ations : 

1.  What  will  represent  a  number  in  which  there  are  z 
hundreds,  y  tens,  and  x  units? 

2.  (a-3)(3+a)  {a'+h^y  %  (s-5)(s-4) 

3.  If  2x-f-l  represents  an  odd  number,  what  will  represent 
the  next  smaller  odd  number? 

4.  (x-3)(x+2)  {a'-¥y  (a+5)(a-3) 

6.  A  has  n  cows,  B  has  5  more  than  A,  and  C  has  as  many 
as  A  and  B  together.     How  many  have  all? 

6.  (4+6)(6-4)  {a'-^¥y      .  (n+9)(w-8) 

7.  What  will  represent  the  sum  of  five  consecutive  even 
numbers  of  which  7n  is  the  middle  one? 

8.  (x-l)(x-^c>)  {a'-¥y  (a+8)(a+2) 

9.  The  perimeter  of  a  square  is  12x  feet.     What  will 
denote  the  number  of  square  feet  in  its  area? 

•10.  (6-5)(5-f6)  (3a+26)2  (h-9)ib-Q) 

11.  At  m  dollars  a  week  for  men  and  b  dollars  a  week  for 
boys,  how  much  will  6  of  each  earn  in  4  weeks? 

12.  (x+7)(x+4)  {4x-xyy  (2/+8)(i/-3) 

13.  How  many  square  yards  are  there  in  the  ceiling  and 
walls  of  a  room  4x  ft.  by  3a;  ft.  and  2/  ft.  high? 

14.  (7+a;)(a:-7)  (5a4-36)2  ^a-S){5+a) 

15.  What  may  represent  the  area  of  any  rectangle  the 
length  of  which  is  8  inches  greater  than  its  width? 

16.  At  a  cents  a  square  yard,  what  will  it  cost  in  dollars 
to  plaster  a  ceiling  I  feet  long  and  w  feet  wide? 


REVIEW  AND  PRACTICE  165 

Exercise  93 
Solve  the  following  problems  and  equations : 

1.  The  difference  of  two  numbers  is  9,  and  their  product  is 
630.     Find  the  numbers. 

2.  6a:2-5x-5=-4.T  3.  6^/2  +  2?/  + 12  =  31/+ 14 

4.  The  sum  of  the  squares  of  three  consecutive  numbers  is 
245.     What  are  the  numbers? 

5.  8.T2+3a--9=-3x  6.  Gj/^+Qy-f  15  =  2i/+13 

7.  The  sum  of  the  squares  of  three  consecutive  even 
numbers  is  308.     Find  the  numbers. 

8.  2a;2-2a:+3=-7a;'    ,  9.  3y''-2y-{-17  =  9y+n 

10.  One  number  is  f  of  another,  and  the  difference  of  their 
squares  is  80.     Find  the  numbers. 

11.  (a;+4)2-9  =  3(3x+9)  12.  ax''-a  =  bx^-b 

13.  The  square  of  a  number  exceeds  the  square  of  f  of  it 
by  567.     What  is  the  number? 

14.  {x-\-S){x+7)=S(x-h5){x-2)+Q 

15.  A  rectangle  of  land  5  times  as  long  as  it  is  wide  con- 
tains 8  acres.     Find  the  dimensions. 

16.  There  are  48  sq.  yd.  in  a  floor  which  is  6  feet  longer 
than  it  is  wide.     Find  the  dimensions. 

17.  The  square  of  a  number  increased  by  the  square  of  half 
the  number  equals  980.     Find  the  number. 

18.  Four  equal  squares  of  paper  contain  208  square  inches 
less  than  one  square  28  inches  on  each  side.  Find  the  length 
of  each  of  the  four  squares. 

19.  A  man  bought  land  for  $1280,  paying  |  as  many 
dollars  per  acre  as  there  were  acres  in  the  piece.  At  what 
price  per  acre  did  he  buy  the  land? 


166  ELEMENTARY   ALGEBRA 

Exercise  94  —  Oral  Practice 

Formulate    the   odd   numbered   exercises    and    give    the 
products  in  the  even  numbered  exercises. 

1.  If  a  rectangle  is  6  in.  longer  than  wide,  what  are  the 
dimensions,  if  each  is  increased  8  in.  ? 

2.  {x+7){x-{-7)  (a+3)(a-l)  {x+4){x-}-2) 

3.  What  is  the  area  of  a  square  formed  by  adding  3  feet 
on  all  sides  of  a  square  x  feet  long? 

4.  (a:-6)(x+5)  (a-2)(a-l)  {x-\-4:){x-S) 

6.  What  may  represent  the  perimeters  of  the  first  and  the 
enlarged  rectangles  in  the  first  problem? 

6.  (a;-7)(x-5)  (a-8)(a-8)  {x-4){x-\-l) 

7.  What  will  represent  the  sum  of  four  consecutive  even 
numbers  of  which  n  is  the  largest? 

8.  (x+6)(x-f 5)  (a-8)(a-f3)  ix-\-Q){x-\-Q) 

9.  What  restriction  is  placed  on  the  exponents  used  in 
proving  the  law  of  exponents  for  multiplication?     (§  132.) 

10.  {x-H)ix-Q)  (a-f8)(a-7)  (?i+8)(n+3) 

11.  What  does  {x-{-2y  represent,  if  x  in  the  expression 
represents  the  side  of  a  square? 

12.  {s-7){s-7)  (6-9)(6+3)  {x-9){x-4) 

13.  Write  5  times  the  square  of  a  —  b,  diminished  by  the 
product  of  the  binomials,  x  —  7  and  re  — 9. 

14.  (n+5)(n-h2)  (a+8)(a+8)  (x-|-7)(x-l) 

16.  What  does  (x— 4)(a;  — 3)  represent,  if  x  in  the  expres- 
sion represents  the  side  of  a  square? 

16.  (s-9)(.s-9)  (6-6)(6+4)  (?/+9)(2/+7) 

17.  What  will  represent  the  quotient  of  a  number  of  three 
figures  divided  by  3  times  the  sum  of  the  digits? 


REVIEW  AND   PRACTICE  167 

Exercise  95  —  Problems  for  Review 
Solve  the  following  problems  and  exercises: 

'  x-4:'^x+4:       2  '48 

3.  The  sum  of  two  numbers  is  24,  and  their  product  is 
128.     Find  the  numbers. 

4.  The  sum  of  the  squares  of  three  consecutive  odd  num- 
bers is  37 1 .     Find  the  numbers. 

6.  The  sum  of  two  even  numbers  is  18,  and  the  sum  of 
their  squares  is  164.     Find  the  numbers. 

6.  Find  two  numbers  whose  difference  is  8  and  whose  sum 
multipUed  by  the  smaller  number  is  280. 

7.  Find  two   consecutive   numbers   the   sum   of  whose 
squares  exceeds  10  times  the  smaller  number  by  155. 

8.  Find  the  side  of  a  square  whose  area  is  doubled  by 
increasing  its  length  6  in.  and  its  width  4  in. 

9.  The  square  of  the  sum  of  two  consecutive  numbers 
exceeds  the  sum  of  their  squares  by  112.     Find  the  numbers. 

10.  A  man  worked  17  times  as  many  days  as  he  received 
dollars  per  day  and  earned  $272.  How  many  days  did  he 
work  and  how  much  did  he  receive  per  day? 

11.  At  20(f  a  square  foot,  it  cost  $56  to  lay  a  parquet 
floor  in  a  room  whose  length  is  6  feet  more  than  its  width. 
Find  the  dimensions  of  the  floor. 

12.  A  mason  worked  32  days  more  than  he  received  dollars 
per  day  for  his  labor  and  earned  $105.  How  many  days  did 
he  work  and  how  much  did  he  receive  per  day? 

13.  An  aeroplane  flew  50  more  miles  an  hour  than  the 
number  of  hours  it  flew.  It  flew  399  miles  on  the  trip  in 
question.     How  long  was  it  in  making  the  trip? 


168  ELEMENTARY  ALGEBRA 

Exercise  96  —  Oral  Review 
Answer  the  questions  and  perfonn  indicated  operations: 

1.  What  does  (x+2)(x  — 2)  represent,  if  x  in  the  expres- 
sion represents  the  side  of  a  square? 

2.  (a;-6)(x-2)  (5a -56)2  (^4_7)(^i_|_5) 

3.  At  X  cents  a  rod,  how  many  dollars  will  it  cost  to  enclose 
a  rectangular  field  I  rods  by  w  rods? 

4.  {x-\-S){x+2)  (a+7)(a-5)  (n-4)(n+2) 

6.  What  is  the  area  of  a  square  formed  by  cutting  off  a 
strip  2  yards  wide  from  all  sides  of  a  square  x  yards  long? 

6.  (a;+9)(x-7)  (a-9)(a-f-5)  (n-8)(n-3) 

7.  What  will  represent  the  quotient  of  a  number  of  x 
hundreds,  y  tens,  and  z  units,  divided  by  8? 

8.  (x+8)(x+4)  (a+8)(a-4)  (n-9)(n-7) 

9.  What  is  received  for  x  sheep  bought  at  a  dollars  a  head 
and  sold  at  a  profit  of  b  dollars  a  head? 

10.  (x+6)(x-h3)  (a-}-5)(a-2)  (/i-9)(?i-f-4) 

11.  A  man  worked  8  days  of  n  hours  each  at  x  cents  an  hour. 
He  spent  b  dollars.     How  much  had  he  left? 

12.  (x-3)(x-l)  (a-5)(a+2)  (?i+8)(n+7) 

13.  What  is  received  for  y  horses  bought  at  p  dollars  a  head 
and  sold  at  a  loss  of  q  dollars  a  head? 

14.  {x+Q){x-2)  (a-8)(a-f  1)  (n-7)(n-6) 

16.  A  rectangular  field  5x  rods  long  has  a  perimeter  of  18a: 
rods.     What  will  denote  the  area  in  acres? 

16.  (x+9)(a:+3)  (a+9)(a-5)  (n-8)(w-l) 

17.  If  the  quotient  is  represented  by  q,  the  divisor  by  d, 
and  the  remainder  by  r,  what  is  the  dividend? 


REVIEW  AND   PRACTICE  169 

Exercise  97  —  Test  Questions 
Answer,  solve,  and  perform  indicated  operations: 

1.  What  is  the  last  step  in  finding  the  root  of  an  equation? 
What  axiom  is  involved? 

2.  (204-2)2  (30-1)2  (40+5)(40-5) 

3.  Define  transposition.     State  the  principles  that  are 
involved  in  transposing  a  term. 

4.  (40+5)2  (50-1)2  (20+3)(20-f2) 

5.  Indicate  the  product  of  three  binomials  without  using 
the  sign  of  multiplication. 

6.  (60+5)2  (40-1)2  (30-6)(30-4) 

7.  Read  the  sum  of  a(x+i/)  and  b(x-^y).     Of  a(b—l)  and 
(h—1).     Of  a(m  —  n)  and  2(m  —  n) . 

8.  (20+8)2  (30-5)2  (40+5)(40-4) 

9.  Define:   power;  square;  cube.     How  do  you  find  the 
square  of  a  number?     The  cube? 

10.  (30+6)2  (50-4)2  (50-7)(50+5) 

11.  State  the  law  of  signs  to  be  observed  in  raising  a  mono- 
mial to  any  power. 

12.  Write  two  identities  that  express  in  algebraic  symbols 
the  rules  for  squaring  any  binomial. 

13.  When  is  the  value  of  a+6  a  negative  number?     When 
is  the  value  of  a  —  6  a  positive  number? 

14.  Indicate  the  product  of  two  binomials  and  two  mono- 
mials without  using  the  sign  of  multiplication. 

16.  Write  an  identity  which  tells  how  to  find  the  product 
of  the  sum  and  difference  of  any  two  numbers. 


170  ELEMENTARY  ALGEBRA 

16.  Read    the    sum    of    (a-\-b){x-\-y)    and    {a—b){x-\-y). 
Read  the  sum  of  (a+  c)  (n  —  1 )  and  ( c — a)  (n  —  1) . 

17.  Write  an  expression  that  represents  5  times  the  square 
of  the  sum  of  any  two  numbers. 

18.  From  what  law  do  we  obtain  the  rule  for  multiplying 
a  polynomial  by  a  monomial? 

19.  Show  that  the  difference  of  the  squares  of  two  consecu- 
tive integers  is  an  odd  number. 

20.  Represent  3  times  the  sum  of  the  squares  of  any  two 
numbers  multiplied  by  their  difference. 

21.  Show  when  the  product  of  several  negative  numbers  is 
positive  and  when  it  is  negative. 

22.  From  4ab  —  3ac-\-2bc  subtract  the  sum  of  Sbc+bd—ac, 
Sab  —  2bd—bc,  and  bd  —  2ac  —  ab. 

23.  What    does    a^+b^    represent?     What    does    x~  —  y- 
represent?     What  does  2(a+l)(a— 1)  represent? 

24*  Define  coefficient;  exponent;  and  show  the  difference  in 
their  meaning  or  signification. 

26.  What  does  2{a-\-hy  represent?    What  does  3(a-6)2 
represent?    What  does  (a + 6)  (a  —  b)  represent? 

26.  Subtract  7x  —  5y-\-3z  from  3x  —  Sy-{-Qz,  subtract  result 
from  zero,  and  add  to  4x  —  Sy-}-2z. 

27.  Simplify    12a-(26- c)+4c-(5a-h36)    and    find    its 
value  when  a  =  7,  6=— 3,  c=  —4. 

28.  State  the  sign  law  of  multiplication.     State  the  index 
law  of  multiplication.     Prove  both  laws. 

29.  Represent  5  times  the  sum  of  the  squares  of  any  two 
numbers  multiplied  by  the  square  of  their  sum. 

30.  How  much  does  the  square  of  70+3  exceed  the  product 
of  (70  4- 3)  (70 -3)?     Give  result  without  squaring. 


REVIEW  AND  PRACTICE  171 

31.  Subtract  the  sum  of  5m—a  —  9n  and  56+5n+a— 4m 
from  a-|-46-|-6m  — 4w. 

32.  Represent  the  product  of  any  three  numbers,  the  last 
two  of  which  differ  by  2. 

33.  How  is  the  dividend  found,  when  the  divisor,  quotient, 
and  remainder  are  known. 

34.  State  the  sigii  law  of  division.     State  the  index  law  of 
division.     Prove  both  laws. 

36.  Without  squaring  the  binomial,  give  the  difference 
between  (60+4)^  and  (60+4)(60-4). 

36.  Find    the    value    of     (a-6)2+(6- c)2+(a-6)-f2c2 
when  a=l,  6  =  3,  and  c=  —4. 

37.  Write  the  product  of  51  and  49  by  expressing  them  as 
the  sum  and  difference  of  two  numbers. 

38.  How  do  you  determine  whether  a  trinomial  of  the  form 
of  x'^-\-bx-{-c  is  the  product  of  two  binomials? 

39.  Represent  4  times  the  sum  of  the  cubes  of  any  two 
numbers  multiplied  by  the  sum  of  their  squares. 

40.  Show  that  the  difference  of  the  squares  of  two  consecu- 
tive odd  numbers  is  twice  the  sum  of  the  numbers. 

41.  Add  (a-f  c)-h2a(6  +  c),  b(b  -  c) -\-a{a-\-c)-{b-\-c), 
{a-\-c)  —  {b  —  c)—a{b-\-c),  and  4(6  — c)  +  (a+c). 

42.  From  the  sum  of  2ab  —  ac-\-2bc  and  2ac  —  bc  —  Sab  sub- 
tract the  sum  of  3ac — 46c  —  ab  and  2a6c  —  2a6  —  2ac. 

43.  Find  the  cost  of  x  books  at  a^  apiece,  x+5  books  at 
b^  apiece,  and  x  — 3  books  at  n^  apiece. 


CHAPTER  XVI 

HIGHEST  COMMON  FACTOR.    LOWEST 
COMMON   MULTIPLE 

HIGHEST  COMMON  FACTOR 

217.  A  common  divisor,  or  common  factor,  of  two  or  more 
numbers  is  an  exact  divisor  of  each  of  them. 

Thus,  a-  is  a  comtnon  factor  of  2a^,  3a*b,  and  a^bc. 

218.  The  highest  common  factor  (h.c.f.)  of  two  or  more 
numbers  is  the  product  of  all  their  common  factors.     Thus, 

x^  is  the  h.c.f.  of  x^,  x'^y,  and  2x^y^z. 

The  term  greatest  common  divisor  is  used  in  arithmetic,  but  it  is  not 
appUcable  in  algebra.  For  example,  x^  above  may  or  may  not  be 
greater  than  x.  Thus,  if  x  =  ^,  x^=^,  and  x^  is  therefore  less  than  x. 
In  algebra  the  term  highest  common  factor  is  used.  That  is,  x^  is  higher 
than  X  (meaning  x^)  in  the  sense  that  its  exponent  is  higher  than  that  of  x. 

fflGHEST   COMMON   FACTOR   OF   MONOMIALS 

219.  The  highest  common  factor  (h,  c.  f.)  of  two  or  more 
monomials  may  be  determined  by  inspection.     Consider: 

Sa^c\    4a26c^,     IGa^ft^c^,     12aV 

The  h.  c.  f.  of  the  coefficients  is  4.  The  highest  common  literal 
factors  are  a^  and  c^.     The  h.  c.  f.  is  4aV. 

Observe  that  the  power  of  each  letter  in  the  h.  c.  f.  is  the  lowest 
power  of  that  letter  found  in  any  of  the  monomials. 

220.  Rule. —  To  the  h.c.f.  of  the  coefficients,  annex  the 
highest  power  of  each  letter  conimon  to  all. 

172 


HIGHEST  COMMON   FACTOR  173 

Exercise  98 
Give  the  h.  c.  f.  of  each  of  the  following  sets  of  numbers: 
1.  a',  a\  2a^b  2.  ^xY,  ^^f,  SxHf,  Qx^ 

3.  (jn\  97^^  San^  4.  Sa^b\  9a^b^,  Qa'^b^  12a^b^ 

6.  4x%  2x\  Sx^ij  6.  5aV,  Sa^x\  2a''x',  lOa^x^ 

7.  W,  3a^  12a^b  8.  2x''y',  SxH/,  Qx^y^,  14xV 
9.  5x^  6^6,  lOax^                 10.  Qa^n\  3aV,  9aV,  ISa^n^ 

fflGHEST   COMMON   FACTOR   OF  POLYNOMIALS   BY   FACTORING 

221.  To  find  the  highest  common  factor  of  compound 
expressions  by  factoring,  proceed  as  follows: 

4a;2-24x+36  =  4(a:-3)(x-3) 
6a;2-42a;+72  =  6(a:-3)(a:-4) 
2x^+12x-5'i  =  2ix-S){x+9) 

The  common  prime  factors  of  these  numbers  are  2  and  x  — 3, 
and  the  h.c.f.  is  their  product,  or  2.t  — 6. 

222.  Rule. —  Resolve  the  nutnbers  into  their  prime  factors, 
and  find  the  product  of  all  the  common  factors. 

Observe  that  each  factor  is  taken  the  least  number  of 
times  it  is  found  in  any  of  the  given  expressions. 

Exercise  99 

Find  the  highest  common  factor  of  the  following  expres- 
sions, by  inspection  as  far  as  possible : 

1.  3x^+3y^,  x-\-y,  and  x*  —  y^ 

2.  2x2-14x+24  and  x^-Qx+9 

3.  5a'^  —  ob*  and  ac-\-ad—bc—bd 

4.  5x^+40,  ^2-4,  and  x^+Gx+S 

6.  3x2-llx-20  and  Sx^-12x-15 


174  ELEMENTARY  ALGEBRA 

6.  x^-Qx+O,  x''+2x-15,  and  a:3-27 

7.  2ax  —  2af  Qax^  —  Qax,   and  2abx  —  2ab 

8.  lHax'^-\-Qa,  lSax^  —  2a,   and  54ax+2a 

9.  27a'-U,   9a2-16,  and  Sa'-2Sa-\-32 

10.  40^2-20x4-25,  8x3-125,  and  4x^-25 

11.  x3+27,  x2-9,  8ax2+24ax,  and  x^-81 

12.  24x^-81x,  12x2- 18x,  and  48x3 -108x 

13.  a2+3a6-1862,  a''-27a¥,  and  (a -36)2 

14.  a'»+a2_2^  Sa^b-Sab,  and  4-8a2+4a^ 
16.  a2-4a6+462,  a^-S¥,  and  a'-ab-2b^ 

16.  9x2-6x+l,  6x2+10x-4,  and  Oax^-ax 

17.  \^a^bc-Wbc,  16-a^  8-a«,  and  4-a^ 

18.  a4-4a262+36S  a'-b\  and  6^-2a2624-a4 

19.  x2-10x+16,  \2xy-Zxhj,  and  x2-4xH-4 

20.  63a2-36a,  49a2-16,  and  16-56a+49a2 

21.  24x2+18x-15,  1-4x4-4x2,  and  8x3-2x 

22.  x2+12x+36,  x2-2x-48,  and  x2-3x-54 

23.  9x2-12ax+4a2,  and  2a6+2ai/-36x-3xi/ 

24.  a2_[-2ac+c2,  a^  —  a?c  —  a&-\-(?,  and  4a2— 4c2 
26.  ay^-\-ax^,  bo?x^-\-ba?xy,  and  ax^-\-ay'^-\-2axy 

26.  64a-32ax+4ax2,  5(x-4)2,  and  x2+2x-24 

27.  ba^-\-ba¥,  aH  —  a}y  —  abx-\-aby,  and  a^  —  abx 

28.  x^  —  x^y—xy^-\-y^,  (x  —  yY,  and  x*  —  2x^y^-{-y^ 

29.  Sa^-Sb^,  a^-^-a'b-ab^-b^,  and  b^-2ab-\-a' 

30.  a^-f-a^c— ac2  — c^,  ax  —  ay  —  cx-\-cy,  and  a2_c2 

31.  16x3+4x2-2x,  2x- 16x2+32x3,  and  IGx^-x 


LOWEST  COMMON   MULTIPLE  175 

LOWEST   COMMON   MULTIPLE 

223.  A  multiple  of  a  number  is  a  number  that  is  exactly 
divisible  by  that  number.     For  example, 

4a6,  8ac,  and  2ax  are  multiples  of  2a. 

224.  A  common  multiple  of  two  or  more  numbers  is  a 
number  that  is  exactly  divisible  by  each  of  them.     Thus, 

\2a%c  is  a  common  multiple  of  2a,  36,  and  2c. 

225.  The  lowest  common  multiple  (1.  c.  m.)  of  two  or  more 
numbers  is  the  product  of  all  their  different  factors.      Thus, 

18a^  is  the  1.  c.  m.  of  3a,  Qa^,  and  6a^. 


>.  Principles. —  Every  multiple  of  a  number  contains 
all  the  factors  of  that  number. 

The  lowest  common  multiple  of  two  or  more  numbers  con- 
tains only  the  factors  of  all  the  numbers. 

If  two  or  more  numbers  have  no  common  factor,  their  lowest 
common  multiple  is  their  product. 

LOWEST   COMMON    MULTIPLE   OF   MONOMIALS 

227.  The  lowest  common  multiple  of  two  or  more  mono- 
mials is  determined  by  inspection.     Consider : 

2abc,        Qab\        Sa%X        4a¥,        a%c 

The  1.  c.  m.  of  the  coefficients  is  12.  The  lowest  common  multiple 
of  the  literal  parts  is  a%^c.    Hence  120^6^0  is  the  lowest  common  multiple. 

Observe  that  the  exponent  of  each  letter  is  the  highest  exponent  that 
letter  has  in  any  one  of  the  monomials. 

228.  Rule.  —  To  the  lowest  common  multiple  of  the  coefficients, 
annex  all  the  letters  of  each  monomial,  giving  each  letter  the 
highest  exponent  it  has  in  any  monomial. 


176  ELEMENTARY  ALGEBRA 

Exercise  100 

Give  the  lowest  common  multiple  of  the  following: 

1.  2a2,  Sa\  5a^b  2.  ^ax"",  2a'x,  5ay\  lOa^x 

3.  Sx\  6i/2,  9x''y  4.  9a^b,  4ab\  Sa^a,  12¥c 

6.  6a^  5x^  3a^x  6.  4x1/,  Sx'^y,  oxy^,  15x^z 

7.  07i\  2n^  S¥n  8.  8a*6,  Sfe^o:,  4aa:3,   l&x^^ 
9.  4a^,  5  c-,  la^h                            10.  5a:^?/,  Ixy^,  2x^y,  \4yz^ 

LOWEST   COMMON    MULTIPLE   OF   POLYNOMIALS  BY  FACTORING 

229.  The  lowest  common  multiple  of  polynomials  is  found 
by  resolving  them  into  their  prime  factors,  and  finding  the 
product  of  all  the  different  factors.     For  example : 

a2+7a+12=(a-f3)(a+4) 
o2+8a+16=(a4-4)(a+4) 
a2-4a-32=(a+4)(a-8) 
The  1.  c.  m.  is  (a+3)  (a  -  8)  (a+4)2. 

230.  Rule. —  Find  the  product  of  all  the  different  prime 
factors  of  the  numbers,  taking  each  factor  as  many  times  as  it  is 
found  in  any  of  the  given  numbers. 

The  factors  of  the  lowest  common  multiple  may  often  be  determined 
without  writing  the  factors  of  the  expressions.     Consider: 

2x-\-y,         2xy  —  y'^,        4x^  —  y- 

The  different  factors  in  these  expressions  are  y,  2x-\-y,  and  2x—y, 
and  the  lowest  common  multiple  is  y{4x^—y-). 

Exercise  101 

Find  the  1.  c.  m.  of  each  of  the  following  exercises,  deter- 
mining it  without  writing  the  factors,  as  far  as  possible: 

1.  x2-3a:-4  and  x^-1 

2.  6a-66  and  4a^-4b^ 


LOWEST  COMMON   MULTIPLE  177 

3.  x^-\-4:X-\-4  and  x'^  —  A 

4.  x^  —  2ax-\-a^  and  a^  — x^ 
6.  2(a2-|-x2)  and  bia*-x*) 

6.  0:^-8  and  x^-lOo^+lG 

7.  x'^-\-2xy-{-y^  and  x'^  —  xf- 

8.  a2-62  and  a2-2a6+62 

9.  a;^-!,  x^-fl,  and  a;^-! 

10.  a^-fl,  a^-\-a,  and  a^— 1 

11.  a;2+14x+40  and  x^-lQ 

12.  3c(c-a)2  and  2a{a^-c^) 

13.  3— 4a;4-a:^  and  a:2+4  — 5.x 

14.  27+a;3,  64-2a:,  and  x^-O 
16.  a2+a- 12  and  a2-a-20 

16.  a^  —  2a^b-\-ab^  and  ax—bx 

17.  a  — a;,  a^  — x^,  and  a^  — x^ 

18.  a2_5a-f-4  and  a^-2a-\-l 

19.  a'*  — x^  a  — x,  and  a^—x^ 

20.  a^  — x^  a^  — a;2,  and  (a  —  x)^ 

21.  a2+6a+8  and  a^+5a+Q 

22.  x2+a;-20  and  12-7a:+x2 

23.  l+4a;2,  4a;2-l,  and  2x-l 

24.  d-  —  ax-\-x'^,  a^-|-r\  and  a+x 

25.  a  —  x,  ar-\-x^  —  2ax,  and  x^  — a^ 

26.  a;2-llx+24  and  x'^-Gx-lG 

27.  a2+2a-15  and  21 -\2a-\-a'- 


178  ELEMENTARY  ALGEBRA 

28.  8^3-64,  4^2-16,  and  6a:- 12 

29.  2a+66,  3a-96,  and  Sa'-27b^ 

30.  a2-4,  a2-4a+4,  and  a^-f  2a3 

31.  a^  —  b^,  a—b,  b-\-a,  and  b  —  c 

32.  x'^-5ax-24:a^  and  x2+8aa:  +  15a2 

33.  x^  —  xy,  x^— 2/^  and  x^-\-xij-]-7f 

34.  a2_i^  2a+2,  3a-3,  and  5a-5 

35.  a^  —  Sab  —  ^b^  and  ax— 4a+&x  — 46 

36.  ac{x  —  y),  2a{x-\-7j),  and  3c(x+?/) 

37.  ^2-1,  l-2a;+x2,  and  H-2x+a;2 

38.  20a -5,  IGa^-l,  2a,  and  12a2+3a 

39.  Aa?c-Wc,  2a2+2a6,  and  3a6-36'' 

40.  l-h2x H-a;2,  l-2x''+x\  and  (l-a;)^ 

41.  x^-A:,  x^-f-4x2+4,  and  4-4x2+a;^ 

42.  a2H-8a4-16,  a^-ie,  and  a2-8a+16 

43.  a;3+2a;2-4a:-8  and  x3-2.r2+4x-8 

44.  x2+?/2,  .T7/-7/2,  x?/4-?/2,  and  x^-f-x?/ 
46.  l+x2+x^  1-X+X2,  and  l+x+x^ 

46.  12x2+12,  2x2-2,  8x+8,  and  4x-4 

47.  2a4(a^+x2),  5a3(a2-x),  and  3a2(a2+x) 

48.  x^  —  x^y-\-xy^  —  y^  and  x^+x'^y—xy^  —  y^ 

49.  a2-a-6,  a2-lla+24,  and  a2-6a-16 

50.  x2-2x-3,  x2+2x-hl,  and  9-6x+x2 

51.  a3-3a2-4a+12,  a2-4,  and  a2-a-6 

52.  x2+7x+10,  x2-4x-45,  and  x2-7x-18 


CHAPTER   XVII 
FRACTIONS 

231.  An  algebraic  fraction  is  the  indicated  division  in 
fractional  form  of  one  number  by  another  (see  §  7).  As 
examples,  observe: 

a-\-b  x-\-y  a^—¥ 

232.  The  numerator  is  the  number  above  the  line.  The 
denominator  is  the  number  below  the  line. 

The  numerator  of  a  fraction  represents  the  dividend,  and 
the  denominator  represents  the  divisor. 

The  numerator  and  denominator  of  any  fraction  taken 
together  are  called  the  terms  of  the  fraction. 

Recall  that  the  dividing  line  is  a  symbol  of  aggregation  as  well  as 
one  of  division.     See  §  152. 

233.  An  integer,  or  integral  number,  is  a  number  no  part  of 
which  is  a  fraction,  as  5,  11,  16. 

A  fraction  of  anything  is  defined  in  arithmetic  as  one  or  more  of  the 
equal  parts  of  it;  but  since  the  terms  of  an  algebraic  fraction  may  be 
any  numbers,  positive  or  negative,  integral  or  fractional,  it  is  quite 
evident  that  the  arithmetical  definition  does  not  accurately  describe 
an  algebraic  fraction. 

The  value  of  any  arithmetical  fraction  is  the  quotient  of  the  numerator 
divided  by  the  denominator.  This  is  true  of  any  algebraic  fraction,  and 
for  this  reason  it  is  defined  as  in  §  231  above. 

A  fraction  whose  numerator  is  a-\-h  and  whose  denominator  is  a  —  h, 
is  read:  a +6  over  a—h,  or  a-{-h  divided  hy  a  —  h. 

179 


180  ELEMENTARY   ALGEBRA 

234.  The  sign  of  a  fraction  is  the  sign  written  before  the 
line  that  separates  the  terms. 

235.  Since  a  fraction  is  an  indicated  division,  by  the  hiw 
of  signs  in  division,  §  158,  the  following  is  true: 

-{-3         3  -3         3 

+9.^9  -9__9 

-3        3  +3        3 

Changing  the  signs  of  both  numerator  and  denominator  does 
not  change  the  sign  of  the  fraction. 

Changing  the  sign  of  either  numerator  or  denominator 
changes  the  sign  of  the  fraction. 

If  either  term  of  a  fraction  is  a  polynomial,  its  sign  is 
changed  hy  changing  the  sign  of  every  term. 

a—b_—a-^b_b  —  a 
x-y     —x-\-y     y  —  x 

236.  Two  principles  are  to  be  observed  when  the  terms  of  a 
fraction  are  expressed  by  their  factors,  viz. 

1 .  Changing  the  sign  of  one  factor  in  numerator  or  denomi- 
nator changes  the  sign  of  the  fraction.     For: 

{a-b){b-c)  _  _{a-b){b-c)  _  _{a-b){c-b) 

ix-y)(y-z)         {x-y)(z-y)  {x-y){y-z) 

This  is  evident,  for  changing  the  sign  of  one  factor  changes 
the  sign  of  that  term  of  the  fraction. 

2.  Changing  the  sign  of  two  factors  in  numerator  or  denomi- 
nator does  not  change  the  sign  of  the  fraction.     For: 

(a-b)ib-c)  _{a-b)ib-c)  _{b-a)ic-b) 
(x-y)(y-z)      (y-x)(z^y)      (x-y)(y-z) 

This  is  true,  for  changing  the  signs  of  two  factors  does  not 
change  the  sign  of  that  term  of  the  fraction. 


FRACTIONS  181 

237.  Reduction  of  fractions  is  the  process  of  changing 
their  form  without  changing  their  value. 

Let  a  and  h  denote  any  two  numbers,  and  m  the  quotient  of  a  divided 
by  h.     Expressing  this  in  an  equation, 

a 

-  =  m 
h 

Since  m  is  the  quotient  of  a  divided  by  6,  and  since  the  dividend 
equals  the  product  of  the  divisor  and  quotient,  a  =  hm,  and  by  the 
multipHcation  axiom,  §15, 

a'n  =  bm'n. 

Dividing  both  members  of  the  last  equation  by  6«n,  and  indicating 
the  division  in  the  first  member,  we  have: 

a-n 

b-n 

By  the  comparison  axiom,  §15, 

a    a-n    .  a   , 

-  = ,  smce  m=—  also. 

b     b-n  b 

a    a-n 
We  multiply  both  terms  of  the  first  member  of  -  =- —  by  n  to  get  the 

0     b-n 

second  member,  and  divide  both  terms  of  the  second  member  by  n  to  get 

the  first  member.     This  being  an  equation,  by  the  multipHcation  and 

division  axioms,  §15,  the  value  of  the  fraction  is  not  changed. 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the  same 
number  does  not  change  the  value  of  the  fraction. 


Exercise  102 

a~\~x 

1.  Change to  an  equivalent  fraction  whose  numerator 

„      ,  a—x 

IS  a^  —  x^. 

2.  Change  • to  an  equivalent  fraction  whose  denomi- 

y-x 

natorisx— 2/. 

X 

3.  Change  — — ^  to  an  equivalent  fraction  whose  numerator 
is  x^  —2x. 


182  ELEMENTARY  ALGEBRA 

x-\-2 

4.  Change to  an  equivalent  fraction  whose  denomi- 

x  —  2 

nator  is  {x  —  2y, 

'V 

6.  Change to  an  equivalent  fraction  whose  denomi- 

y-x 

hator  is  a;2  — 1/2. 

238.  A  fraction  is  in  its  lowest  terms  when  the  numerator 
and  denominator  have  no  common  factor  except  1. 

To  reduce  a  fraction  to  its  lowest  terms,  we  must  remove  all  factors 
found  in  both  numerator  and  denominator. 

This  is  done  by  canceling  the  common  factors,  which  is  equivalent 
to  dividing  both  numerator  and  denominator  by  them,  thus, 

15aH''_Sa  x^-3x+2 _Cx^^)ix-l)  _x-l 

239.  Rule. —  Resolve  numerator  and  denominator  into  their 
prime  factors  and  cancel  {divide  out)  all  factors  common  to  both. 

When  the  numerator  of  a  fraction  is  a  factor  of  the  denomi- 
nator, the  numerator  of  the  result  is  1 .     For  example, 
a-\-x  _     1 
a^—x^    a  —  x 

It  is  often  advisable  to  change  the  sign  of  a  factor  in 
one  term  to  make  it  like  a  factor  in  the  other.     Thus : 

(x+7){x-4)_     {x+7)(x-4)_     x+7     ^ 

5i4-x)  5(x-4)  5 

We  change  the  sign  of  the  factor,  4— x,  in  the  denominator  and  also 
the  sign  of  the  fraction,  and  then  cancel  the  common  factor. 

Exercise  103 

Reduce  the  following  fractions  to  their  lowest  terms, 
giving  results  at  sight  as  far  as  possible : 

,    2a3            ^    3x2                 4^^.                 6ct2            ^    12xhj 
1.  —  2.  —  3.  4.  —  6. 

8a2  6X3  g^y  3ct4  lQ^y2 


6. 


9x' 


n.  '-^ 


7.  — 


12. 


8x^ 


FRACTIONS 


8. 


13. 


9yz 
7a6 
Ihc 


14. 


8x3 
4x^ 
9n5 


10. 


16. 


183 

lOo^fe 
\ba¥ 

2Sxy^ 


Reduce  to  lowest 
possible : 
^     a*x-}-Max 


Exercise  104 
terms,  giving  results  at  sight  as  far  as 


a2-4a+16 

4. 

a^+a 

3b-\-Sab 

7. 

9x^y-Sx^y 

x2-8x+15 

10. 

X2-1 

5xy+5y 

13. 

x^-y' 

x'-2xy+y' 

16. 

2a2-4a 

Sab-Qb 

19. 

a^-¥ 

a2+2a6+62 

22. 

4xyH-4 

5xY-5 

26. 

ax"  —  a^ 

a2-2ax+x2 

28. 

9a3-6a6 

6a26-462 

<»i 

a^-x^ 

2. 

¥-1 
¥-1 

6. 

c?-x^ 

a^-x^ 

8. 

a  —  x 
a^-x^ 

11. 

x'-y' 
x'-y' 

14. 

a2+x2 
a6+x6 

17. 

a2-62 

(a+6)2 

20. 

x'-y' 

{x-yf 

23. 

a3+x3 

(a+x)3 

26. 

x'-y' 

{x'-y'f 

29. 

{a+by 

(a'-b^y 

90 

{x'-y'Y 

(f-2ax-{-x^ 


(x+yy 


3. 


6. 


9. 


12. 


16. 


18. 


21. 


24. 


27. 


30. 


33. 


^2-9 


x2-6x+9 
n-1 


n' 


a^-\-4  —  4a 
a2+6-5a 


a^ 


1 


r2_ 


3a+2 


n2+l 


.2_ 


2n+l 


y'-i 

a^+a  — 6 
a2+6a+9 

n2-l 


x^-m 

x^+2x-S 


184  ELEMENTARY   ALGEBRA 

'  Sxy'-Qy'  '  {a'-x^y  '  a'-^l 

„       aH-x^  {a-hy  a'+2+Sa 

'  a^-\-2ax+x^  '  (a'^-b^y  '  a'+3+Qa 

240.  A  mixed  number  is  a  number  one  part  of  which  is 
integral  and  the  other  part  fractional,  as 

x—y  x—3 

241.  A  proper  fraction   is  a  fraction   which   cannot  be 
reduced  to  a  whole  or  a  mixed  number,  as 

x-\-y  abc  x—S 

a+6  xyz  2/— 4 

242.  An  improper  fraction  is  a  fraction  which  can  be 
reduced  to  a  whole  or  a  mixed  number,  as 

a^-¥  x^-5x-\-9  x^-Y 

a2+62  x-2  '  x^-f 


REDUCTION   OF  IMPROPER  FRACTIONS 

243.  An  improper  fraction  is  reduced  to  a  whole  or  a  mixed 
number    by  performing  the  indicated   division.     Thus,   to 

reduce  to  a  mixed  number,  proceed  as  follows: 

a+2 


a^+x        1 .7-1-2 


c^'+2q'      a2-2a-f4 
-2a2+x 
-2a2-4a 

4a+a; 
4a+8 
x-% 

Therefore,        ^5!+^  =  a^-  2a+4-f  ^ 
a-\-2  a+2 


FRACTIONS  185 

We  continue  the  division  until  the  remainder  is  of  a  lower  degree  in 
the  leading  letter  than  the  denominator.  When  the  sign  of  the  first 
term  of  the  remainder  is  plus,  we  write  it  over  the  divisor  at  the  right 
of  the  integral  part,  connecting  the  integral  and  the  fractional  parts 
with  the  plus  sign. 

An  improper  fraction  reduces  to  an  integral  expression  when  the 
numerator  is  exactly  divisible  by  the  denominator. 

244.  Again,  reduce  to  a  mixed  number: 
W-^a''-(Sa-\-n 
3a2-2 
Observe  carefully: 

6a3-9a2-6a+ll|3a2-2 
6a^  -4a  2a-3 

-9a2-2a+ll 
-9a2___f_6 
-2a+  5 

Therefore     6a3-9a^-6a+ll_  2a-5 

iheretore,  ^^^_^  -la     6     ^^^_^ 

When  the  sign  before  the  first  term  of  the  remainder  is  minus,  change 
the  sign  of  each  term  of  it,  write  it  over  the  divisor,  and  annex  it  to  the 
integral  part,  connecting  parts  of  the  quotient  with  the  minus  sign. 

Why  connect  them  with  the  minus  sign? 


Exercise  106 
Reduce  to  whole,  or  mixed  numbers : 


1. 

o3+l 

a 

A 

x^-l 

x^ 

7. 

a2-l 

2.  ?^  3.  " 


x+1  a  —  b 

a^  — 4  ^    x^  —  y^ 

5.  6.  

a  — 2  x-{-y 

^2-4  a4+.T^ 

8.  9.  

a  — 1  x+2  a  —  x 


186  ELEMENTARY   ALGEBRA 

,o.V^  n.^!±i  12.^^+^^ 


n+1  a  — 2  x-\-y 

13.^!^  ii.'f±l  16.  ?!z^ 

a  — 1  n+2  a;  — 2/ 

16    ^'+^  17    ^'-^  18    ^'-^' 

.      '''  5-x '°--T+2- 

„^    12a2-4a+5  „„    fe2-76+12 

21.  22.  

2a  6-3 


REDUCTION   OF   MIXED   EXPRESSIONS 

245.  Mixed  expressions  are  reduced  to  improper  fractions 
as  in  arithmetic,  except  that  when  the  fractional  part  is 
minus,  the  numerator  of  it  is  subtracted.     Observe : 

a2+9  (a-3)(a+2)=   a^-a-Q 


^    ^~^a+2  Adding  a^       +9 

Hence,  a  — 34 


2a2-a+3 
a2+9     2a2-a-f3 


a+2  a+2 

.,  a^-^-x^  {a  —  x){a  —  x)=a'^  —  2ax-\-x'^ 

a  —  x  Subtracting  a^  -{-x^ 


—  2ax 


Tx  a^+x^     —2ax  2ax 

Hence,  a  —  x— = or 

a  —  x      a  —  x  a—x 

Exercise  106 
Reduce  to  improper  fractions : 

,  ^  ,  2aH-5  ^    8a  — 2a;    „    ,  ^ 

1.  a+lH — r—  2.  — - — -2a+3j 

4a  3 


FRACTIONS 


187 


3.  x-3- 

5.  a-4  + 


Sx-4: 
2x 

4a+3 


7.  a+5- 


9.  x-2 


ba 

7a+4 


a+5 
4x  — 5 


11.  a-44 
13.  x+6- 
16.  a+6- 


a-16 


a+4 
a;-36 

aH-262 
a+6 


4.  ?4^-3x-2. 


6.  5a-36 


8a -46 


8.  'S^-2.-Zy 


Sx-2ij 


10.  2a -4a;- 


Sa'-dx^ 
4a+3a; 


12.  6^^!z9^-3x-4, 


14. 


2x-Sy 

4x^+92/^ 
3a;-2?/ 


16.  3a -2a;- 


•4X+32/ 


8a^-7a;^ 
2a -3x 


LOWEST   COMMON   DENOMINATOR 


246.  Two  or  more  fractions  have  a  common  denominator 
when  their  denominators  are  the  same  numbers. 

The  lowest  common  denominator  (Led.)  of  two  or 
more  fractions  is  tlie  l.c.m.  of  their  denominators. 


Consider: 


a 


a(a-\-x) 


a  —  x     {a—x){a-\-x) 
a  a(a  —  x) 


and 


a-\-x     {a-{-x){a—x) 

247.  Rule. —  Find  the  lowest  common  multiple  of  the  denom- 
inators for  the  lowest  common  denominator. 

Divide  this  denominator  by  the  denominator  of  each  fraction 
and  multiply  both  terms  of  the  given  fraction  by  the  quotient. 


188  ELEMENTARY  ALGEBRA 

Exercise  107 

Reduce   the  following  fractions  to  equivalent  fractions 
having  the  lowest  common  denominator : 


1. 

3a2 
3' 

2ax 
2  ' 

4:xy 
6 

3. 

2a  c 
h  ' 

4^2 

a 

bax 
c 

6. 

4 

2ah' 

3 
6a^' 

b 
^ax 

7. 

3a2 
3' 

4a6 

X 

2xy 
c 

9. 

a 
2bx 

4 

5a2 

Sab 
'     2 

11. 

5  ' 

a 

4bx 
a 

13. 

3 
Sax 

a 
662' 

c 
4ab 

2. 

x+1 
a 

x-\ 
b   ' 

X2-1 

c 

4. 

2 

a-1' 

3 

a+l' 

4 
a2-l 

6. 

a2+4 
a 

aH-2 
6    ' 

a-5 

2 

8. 

b 

c 
x-S' 

a 
x+S 

10. 

5 
4-^2' 

3 

2-:.' 

4 

2+x 

12. 

0:2-4 

i 
a 

a-1 
b    ' 

6-2 
c 

14. 

a2+4 
a2-4' 

a-2 

a+2' 

a+2 
a-2 

ADDITION   AND    SUBTRACTION    OF   FRACTIONS 

248.  Similar  fractions  are  added  or  subtracted  by  perform- 
ing those  operations  upon  the  numerators  and  writing  the 
result  over  the  common  denominator. 

We  have  learned  in  division  that: 

a-{-c-\-e—n  —  x_acc    n    x 
6  6'^6"^6~6~6 

Interchanging  the  members  of  this  equation,  we  observe 
the  rule  for  addition  and  subtraction  of  fractions. 

249.  Rule. —  Reduce  the  fractions  to  the  Led.,  change  the 
signs  of  all  the  terms  of  numerators  of  fractions  that  are  pre- 


FRACTIONS  189 

ceded  hy  the  minus  sign,  find  the  algebraic  sum  of  the  numer- 
ators, and  write  it  over  the  least  common  denominator. 

Observe  the  following : 

a  a        a'^  —  axa^-\-ax       2a^ 


a-\-x     a—x     a-—X"     a'^—x'^     a^—x'^ 

In  many  examples  in  addition  and  subtraction  it  is  best  to  express  the 
lowest  common  denominator  in  its  factored  form. 


Exercise  108 
Perform  the  indicated  operations : 

2a-{-x     a  —  3x  2a-\-h     h-{-c     a-^c 

1.  — 2. -\- 

3            4  3           2          2 

da-\-b     a-2b  ^    Zx-{-ij_x-y     x-j-y 

a             b  '428 

Aa  —  x     2a-\-x  Aa-\-x     a  —  x     x  —  a 

5. 6.  -f- 

3  5  2           6          3 

'  \-\-2x     l-3a;  '3           2          4 

9.-A._^  10.^-^  +  -^ 

2x+3     2a:+8  a+x    a'-x'    a^x 

11.  3a+4^5-3ff  ^^        X     ^    y    _x-y 

4  5                •  x^  — 2/2    x+y    x-\-y 
2x+3     2-f4a;  x+lO     a:+9     a;+4 

13.    14. \~ 

3             6  2            4           3 

_       2a           2a  ..6           8     ,      2a; 

15.    —  16.  — h 


2a-\-b     2a— b  a-\-x     a  —  x     a^  —  x 

.„    3x+?/     3x+?/  a^x        a     ,     2a^ 

17.    — lo. 


a—x    a-\-x    a^—x^ 


190  ELEMENTARY  ALGEBRA 

19.  -^—  -  -^  20. 

2a- 6     2a+6 

21.  ^IZI-'^J^  22. 

23.  3M:_^+_A_  24. 

a  — 6     3a+6 

„^    4a-l  ,  6a+2 

26. 1 26. 

2a+2    3+3a'  a-h    a+6     h'-o? 

x-\-y       X       x{x'^  —  y) 


X 

--] 

y 

h ^ 

xy 

X 

1    '^ 

f 

x-y 

a;+?/ 

x'-f 

5 

,     3 

9x 

x+4 

'x-4 

x'^-lQ 

a 

6 

62 

27. 


28. 


2/       a:+?/    y(x^-y^) 
b  ¥  d'b 


a+6     (a+6)2     (a+6)3 


29.  -i^-+-^4-     ^^ 


(a— x)^     a  —  x     {a  —  xY 

2  4  2 

(a;-i/)2    7/2-x2     {x^-ij)' 

3  4,2 

31.  -T ;^ ^-^-1- 


a{a-2)     a2-4     a(a+2) 

„^      4a6    ,     2  a+6 

32.  7^ A 


W-a^    a-h    a2H-a6+62 

rt+4        7t"+31        n-2 
^^*  ii^    n2_2^_i5"^n+3 

34. 


y^-\-x^    x-\-y    x^  —  xy-^-y^ 
x-y  {x-zY      ^x-z 

35.  —  7 TT -i 

x—z     {x—z){x  —  y)     x-y 

a+4        a2_^4a-2 1_ 

•  a2+a+l         l-a»        a-1 


FRACTIONS  191 


2  3n+6     ,  n2+3n+5 

gy _L_ 

n+1     n^— n+1         n^  +  l 
^^    a2+4a-f9_    2  a+4 


39. 
40. 


a3+27        a+3     a^-Sa+Q 
n2H-2w+28  ,     3  n+6 


8-n3       'w-2    n2+2n+4 

4a;^     ,       1      ,        x-2y 
x^+S?/^     a:+2|/     x^  — 2aj?/+42/^ 


MULTIPLICATION  OF  FRACTIONS 

250.  The  product  of  two  fractions  is  the  product  of  the 

numerators  over  the  product  of  the  denominators. 

a  c  a     c 

From  T  =  ^  and -:= n,  we  have  YX-;  =  mw.     Why? 
0  a  0    a 

From  the  first  equations,  a=bm  arid  c  =  dn.     Multiplying 

a  =  6m  by  c  =  dn,  member  by  member,  we  have 

ac=bdmn. 

Dividing  both  members  of  ac  =  bdmn  by  hd,  we  have 

—  =  mn,  and  by  the  comparison  axiom, 

a     c_ac 
b    d~hd 
This  method   is   applicable   also   when   either   factor   is 
integral,  for  integers  may  be  expressed  in  fractional  form. 
Since  the  product  of  the  numerators  is  divided  by  the  product  of  the 
denominators,  cancellation  may  be  employed. 

Exercise  109 

SimpHfy  the  following: 

^    U2b^bac  ^    a2-25^a2-9 

!•  ": — X";r~X-r — r  2.  —z — ^r— X- 


Ax     5a    6a6  a'^  —  3a     a'^-\-5a 

Sb    Sa    3cy  a-1       4a^+a 

9a     2x    4ab  '  2a-\-Sa^      a-2 


192  .  ELEMENTARY   ALGEBRA 

6.  ^-1  y}+y  Q    x^-^y\,    x-y 


1  —  y^    x—1  '  x^  —  xy     x'^+2xy 


x^-4:     x-l  '    y^-\-xy      x'^-Qz'^ 

9.    ^'-y'y^^+^  10       x'+8y'^^(x+yy 

x^-\-xz    x  —  y  '    (a;2  — 2/2)2      x-\-2y 


(a+6)2     a-b  '  (x+d)^    yz^+Sy'' 

bc-{-bx     Sax     c  —  x     a^  —  x"^ 
13.      r— X"ri — X X 


a^-{-ax     4:by     a  —  x     c^  —  x^ 


6a2  a  —  x     5by     {a-\-xy 

a-\-b   v,3ac       a  —  b       ab—b 
1^»  ~/        TTTiX    -.    Xt — : — ttitX 


(a-^;)2''   6       (a+^)'     a6+a2 

Ig^  {x-yy^^  x-4:  ^^iy+xy^^x'+27 
d-\-x       0:2  —  1/2     0:^  —  64      x^  —  y^ 

^„     a2_^a6^^  a6-6\  4a;+rr\  a2_52 
17.  7 ttttXt — r-^X — ^ X 


(a- 6)2  ^{a+by      a-\-x        a¥x 

a^-5a-\-Q        a''+2a       a^-^S-6a 
*  a2-2a-8     a2-4a+4     a2+3-4a 
n2+2n+l  n-5  n^-Sn 

19'   ";; :; ;:: — X    „  . — : — r~7: — X 


n^— n2  — 2n    n^-\-4:-\-2n    n^  —  4n  —  5 

ax-\-a'^^a^-\-x'^  —  ax^^Zab     4a2  — Qa;^ 

20. X X X 

4a  — 6a;      2a^  —  Sa^x     2xy      x^-\-a^ 


b     aj     \       aj  \       \  —  aJ\       \-\-a 


y    xj     \       xj  \       1— x/\       1+x 

26.  (x-?^^xf:.-?^^  26.  ("2+   2y  Y,     a.-j/\ 


yj  \      x-yj\      x-\-yJ 


FRACTIONS  -         193 


x-\-zy\        xj  \      n-\-xJ\      n  —  x 


6c/     \       3c/  \       6+c/V       6 

31.  (,,^\x(^-  +  l.)        32.   r       .■^''-^^,       Yl+i+^ 


\2x2+i/v      \2/     2x/  *  \xz-^xy-{-yz/\x    y     z. 

DIVISION   OF  FRACTIONS 

251.  The  reciprocal  of  a  is  1  divided  by  a,  or  -.     The 
reciprocal  of  a  fraction  is  the  fraction  inverted. 

The  reciprocal  of  y  is  -. 
0      a 

252.  The  quotient  of  two  fractions  is  the  product  of  the 

dividend  and  the  reciprocal  of  the  divisor. 

T^         a  ,  c  ,         a     c     m  ,      J. 

From  -r=w  and  -:=n,  we  have  t"^-7  =  — >  by  div.  axiom. 
b  d  b    d     n  . 

From  the  first  equations,   a  =  b7n  and   c  =  dn.     Dividing 

a  =  bm  by  c  =  dn,  member  by  member,  we  have: 

a  _bm 

c     dn' 

Multiplying  both  members  by  -,  we  have : 

ad    m        ,  ,     ^, 

7-  =— ,  and  by  the  comparison  axiom, 

be     n 

a  ^  c  _ad 
b  '  d     be 

253.  Write   integral  expressions   in   the  fractional  form, 
invert  the  divisor,  and  multiply. 

Exercise  110 

Perform  the  indicated  divisions: 

2+a_^4-a2  4x'^-\-x^    2x-{-Sx'^ 

'  x-S '  x-3  x-S  x-2 


194  ELEMENTARY  ALGEBRA 

a-\-S^a'^  —  9  a-\-x    _^  a^-{-ax 

'  '  ■  '  *  ~     ~  '  a^-Sax  '  a2-9x2 

a^-W  '  2ab-\-a^ 

a—2x   '   a^  — 8x^ 

(x2-25)2  ,  (x-5)2 


7. 


6+1* 

62-1 

^+?/  . 

x^^xy 

a+x 

a'-x^ 

a-h 

(a -by 

a+6  * 

a^-b^ 

a2-25 

1     a2-5a 

• 

6. 


9. -^ 10. 

a2+6a     a2_36 


2/     2//      \       X 
17.  Il+-Vfl+?)  18. 


Sa^+ax^  * 

a:+2a 

x2+4-2x 

.x2+8 

x-2 

'a;2-4 

a^  —  x^ 

a  — a: 

x^-Zx-^ 

'x^+o; 

n^  — 6— n  . 

2/1 +n2 

n^-2-n  ' 

n2-2n 

a^+x^ 

a+x 

x/     V       3/  a2+6a+8     4+2a 

x/      \       2//  a^+a;^     ar  —  ax-j-x^ 

21.  (  1--Vfl+-^  22.  ^'+2a;2/+?/^  .  x^-i/^ 


aV     \       a;/  xy-^rlf  'f 

23.     a H-    1+-  24. 


a/     \       a/  a^  —  9  a+3 

26.   (I-i-V^^+ll  26.         ^'-^'        -^'-^^ 


2/2    xv     \x      J                     x^-\-2xy-\-\f-      x-{-y 
27.  U'-^  H-    1--  28. ^^^ 


xj\xj  a^  —  25         5a-\-a'^ 

29.  (  ,.+_J^^l+-j  30.  -^-^-.-^^^^ 


FRACTIONS  195 


31.  (^-4)-.(l-M^  32.^4-    6   ^V.,8-2a 


y^    x^J     \      xj  \      a+1/      \       o?—\ 


Exercise  111  —  Test  Questions  and  Review 
Answer  all  you  can  orally : 

1.  Show  that  a  common  divisor  of  two  numbers  is  a 
divisor  of  any  multiple  of  either  of  them. 

2.  Write  an  algebraic  expression  that  is  exactly  divisible 
by  aH-6  and  2a  — 36. 

3.  Give  some  algebraic  expressions  of  which  the  lowest 
common  multiple  is  2a^  —  2x^. 

4.  How  do  you  determine  whether  a  binomial  is  the  pro- 
duct of  the  sum  and  difference  of  two  numbers? 

6.  Show  that  a  common  divisor  of  two  numbers  is  a 
divisor  of  their  sum  and  also  of  their  difference. 

6.- Recalling  the  solution  of  equations  by  factoring,  what 
are  the  roots  of  the  equation,  a;^  —  5x + 6  =  0? 

7.  Show  how  much  the  square  of  the  sum  of  two  numbers 
exceeds  the  product  of  their  sum  and  difference. 

8.  Without  squaring  the  binomial,  give  the  difference 
between  (30+7)2  ^^d  (30 +7)  (30 -7). 

9.  How  much  does  the  square  of  40 +5  exceed  the  product 

of  (40 +5)  (40 -5)? 

Find  the  value  of  the  following  expressions  when  a  =  l, 
6  =  2,  c  =  3,  d  =  4,  e  =  0,  m  =  \^n  =  \. 

10.  cd2m-8a263-j-d2m2+9c2X262+6a3(/m2n-62c2(i2n 

11.  m3X(i'+76cm-h9a3c3-a2cV+c2(i2^64m+a^Xc^3 


196  ELEMENTARY  ALGEBRA 

Find  the  value  of  the  following  expressions  when  a  =  J, 
b  =  i,  c=l,d  =  4,x  =  ^^y  =  S. 

12.  Qa^d^-5(^y''XS¥x  13.  86y+a(2d2-2d)+5x2 

14.  9¥d^+Sa^d'^5c'x^  15.  2c'd'-{2y'-5d)x-4y^ 

16.  Show  how  much  the  square  of  the  sum  of  two  numbers 
exceeds  the  square  of  their  difference. 

17.  Give  the  difference  between   (20+4)2  and   (20-4)2 
without  squaring  either  binomial. 

18.  What  is  meant  when  it  is  said  that  a  certain  number 
satisfies  an  equation? 

19.  By  what  niust  a  fraction  be  multiplied  to  give  the 
smallest  possible  integral  product? 

20.  Show  to  what  the  sum  of  any  two  numbers  divided  by 
the  sum  of  their  reciprocals  is  equal. 

21.  Find   the    h.  c.f.    and    the    1.  cm.    of    x*-{-x'^y--\-y^, 
x^-\-xy-\-y^,  and  x'^  —  xy-\-y^. 

22.  How  much  does  the  square  of  50+4  exceed  the  square 
of  50—4?     Give  result  without  squaring. 

23.  What  is  the  result  in  multiplication  of  fractions,  when 
all  factors  in  numerator  and  denominator  cancel? 

Find  the  value  of  the  following  expressions  when  a  =  2, 
b  =  l^c  =  4:,d  =  S,n  =  5,x  =  l,y  =  ^. 


d+n 

b^+y 

27.  '+1 

c-b 

y"    U^ 

on      2       2 

30.  — - 

X    y 

31    ^'-^' 

32     ^"'4-^'^ 

33. 

a+2 

b- 

-y 

36. 

X 

y 

a 

X 

FRACTIONS  197 

a^+n'  3n2_8fe2 

'  a^+h^  .       •  (Px     2y^ 

39.  Find    the    h.c.f.    and    the    l.c.m.    of    x^  —  x^  —  x-\-l, 
2x^-x^-x^,  and  2x''+x-S. 

40.  Write  an  expression  of  3  unlike  terms  of  the  third 
degree,  the  terms  involving  x  and  y. 

41.  Define   elimination.     What   axioms   are   involved   in 
elimination  by  addition  and  subtraction? 

42.  By  what  is  the  sum  of  the  same  odd  powers  of  two 
numbers  divisible?     The  difference?     - 

43.  Give  the   difference  between   (60+2)2  ^nd   (60-2)2 
without  squaring  either  binomial. 

44.  State  the  law  for  changing  the  signs  of  one  or  more 
factors  in  numerator  or  denominator. 

45.  How  do  you  determine  whether  a  trinomial  of  the 
form  ax^-\-bx-\-c  is  the  product  of  two  binomials? 

46.  Define   determinate   equation;   indeterminate   equation; 
independent  equations;  simultaneous  equations. 

47.  If  two  equal  fractions  have  the  same  numerator  or 
denominator,  except  0,  how  do  the  other  terms  compare? 

48.  How  much  does  the  square  of  80+2  exceed  the  square 
of  80  —  2?     Give  result  without  squaring. 

49.  If   one   of   the   factors   of   Qa'^x'^—4:aH  —  4:ax^-\-x*-{-a'^ 
is  x2+a2  — 2ax,  what  is  the  other  factor? 

50.  Give  the  two  formulas  that  express  in  general  numbers 
the  index  laws  of  multiplication  and  division. 


CHAPTER   XVIII 

LITERAL  AND  FRACTIONAL  EQUATIONS. 
SOLUTION  OF  FORMULAS 

LITERAL  AND  FRACTIONAL  EQUATIONS 

254.  A  literal  equation  is  an  equation  in  which  there  are 
two  or  more  general  numbers. 

In  solving  such  equations,  the  value  of  any  letter  may  be 
found,  but  only  in  terms  of  the  other  letters. 

Solve  for  x,  ax  —  ar  =  bx  —  6^ 

Adding  a^  and  —bxto  both  members  of  this  equation,  and 
uniting  the  terms  containing  x,  we  have 

{a  —  h)x  =  o?  —  ¥ 
By  the  division  axiom,        x  =  a-\-h 
Checking:       a{a-\-h)  —  a^  =  h{a-\-h) —h"^  or  ah  =  ah 
To  solve  a  literal  equation  for  any  letter  in  it  is  to  find 
the  value  of  that  letter  in  terms  of  the  others. 

Exercise  112 

Solve  the  following  equations  in  the  left  column  for  x,  and 
those  in  the  right  column  for  y  and  check : 

1.  4a-x  =  4b-bx  2.  2b-\-6y  =  Sc+ay 

3.  5n—x  =  4n-\-nx  4.  ay  —  ab  =  3y  —  3b 

5.  3a  —  x  =  2a—bx  6.  5a  — by  =  ay  — 5b 

7.  n^—nx  =  ax—a^  8.  2a  — Qy  =  ay— IS 

9.  _-4a  =  46-^  10.  --n+-  =  y---- 


LITERAL  AND   FRACTIONAL  EQUATIONS  199 

255.  Special  Devices.     It  will  be  well  to  note  here  some 
special   devices   for  clearing  equations  of  fractions.     Thus, 

clear  of  fractions      — - — + =  — — -— 

5         'Sx—1        10 

Multiplying  both  members  by   10,   the  lowest  common 
multiple  of  the  monomial  denominators,  we  have 

3a;  — 1 
Subtract  4x  in  each  member,  unite  other  monomials,  clear 
of  fractions,  and  complete  the  solution. 

In  some  examples  it  simplifies  the  solution  to  combine 
fractions  before  clearing  of  fractions.     Thus,  from 

1      ,  a—c        1      .a-\-c  .  . 


we  obtain 


a—c       X       a-\-c       X 
1  1     _a+c     a—c 

'         a—c    a-hc       x  x 


Combining  the  fractions  in  each  member,  we  have 
2c     _2c 

If  two  equal  fractions  have  the  same  numerator,  not  0, 
their  denominators  are  equal.    Hence, 

x  =  a^—c^ 
Check  by  substituting  in  equation  (1) 

Exercise  113 

Solve  the  following  equations: 

3x+8_4a;-3    x  4y+Sy-2_y+2 

'      12       3x+4    4  *  2/'-4     2+2/     y-2 

2x-5    x_5x+S  .                      3j/-f4_i/+3_l-y 

*  3x-2     3        15  •    1-1/2     i_y    2/+1 
5x—4:_Sx-^S    X  a-\-l  _5x  —  a    a  —  1 

•  ~10       2x+5    2  •  a^~~oF^~^a+l 


200 


7.  —  =  x  —  a-\— 
a  a 


ELEMENTARY  ALGEBRA 
3x  .  X 


«•  ro+r^o-3 


„    3a;     2a; 

9.  ^-^  =  x+3 


11. 


.13. 


15. 


17. 


19. 


21. 


23. 


2+l£5    1^+2 
a6  _  ca; 

ex  a6 

5a^+4_5a:— 4_a; 

2x+2~2 


10. 


12. 


x—c     (2x-— c)' 


X  —  a     {2x  —  ay 
a-}-c    x  —  2a  .  : 


ac 


10 

2x+l 


14.  ?^+26-x^^ 


2x+a    46H-X 


3x-2     6x-l 


5        6x4-3        15 
3x+2     5x4-6  ,  X 


1 


x-6  "^3 
4  5 


X  — 5    x4-l     X  — 2 

5x+4_  6x4-4  _x4-6 
6        3x4-2       5~ 

3x4-8    4x4-8     2x4-5 


6 


3x4-6        4 

2x4-1 


16. 


18. 


20. 


22. 


24. 


X.-2 


c 

x  —  2ac 


2x     ^  ,  6c 

—  =  54- — 
a  a 

l_x-Sb 

X 


bx        X      acx 
8x4-3     x4-2    x+3 


34-x    x-3 
9  8 


x2-9 

6  ^ 

x4-2  2x4-4    8x+3 

2-x  2x.24-2x     x+2 


26. 


26. 


27. 


28. 


29. 


3x2 


x4-2 
2x-l 


2-x 


2x-l     4x2-1     14- 2x 

12x4-ll_9x4-7_10x-5 
8  6        46x4-8 

x4-4a4-c  ,  4x4-a4-2c     _ 


x4-a4-c 
2  3 


x-\-a—  c 
4  5 


X— 2    X— 3     X— 4 
1 


X  — 5 

1 


(x-7)(x4-2)     (x-4)(x-3) 


LITERAL  AND   FRACTIONAL  EQUATIONS  201 

SPECIAL   METHODS 

256.  Observe  the  form  of  the  following  equation  and  the 
method  of  solving  it. 

x-\-2    x  —  5_x  —  Qx-\-S  ,^v 

^+3    x^~x^    x+i  ^  ^ 

Transpose  one  fraction  from  each  side  to  make  each  mem- 
ber the  difference  of  two  fractions.     Thus, 
x-\-2    x-\-3  _x  —  Q    x  —  5 
x-\-3    x-\-4:    x  —  5    x—4: 
Performing  the  indicated   subtraction   in   each   member 
and  simplifying  the  numerators,  we  have 
-1        _        -1 
x2-h7x+12~a:2-9:c+20 
Since  the  numerators  of  these  equal  fractions  are  equal,  and 
not  0,  the  denominators  are  equal. 

x^-9x-\-20  =  x^+7x-\-12 

X  =  -2 

Check  by  substituting  in  (1) 

257.  In  solving  equations  like  the  following,  it  is  best  to 
reduce  the  fractions  to  mixed  expressions: 

5x-7     2a;-17_4x-l     3a;-21  .  . 

~^^     x-1  ~  x-i     x-Q»  ■     y) 


x-2  x-1  x-l  a;-6 

Subtracting  the  sum  of  the  integers  from  both  members, 
dividing  by  3,  and  combining  the  fractions  in  each  member, 
we  have 

-5        _       -5 
x2-9x+14    a;2-7a:+6 
The  denominators  are  equal,  hence, 

a;2-7a:+6  =  a:2-9a;4-14 
a:  =  4 
Check  by  substituting  in  (1) 


202  ELEMENTARY   ALGEBRA 

Exercise  114 

Solve  the  following  equations : 

x-i-2    x+4_2x^-Sx-^2 
'  x-2    x-S~  x^-5x-\-6 

x  —  4iX  —  8_x  —  7x  —  5 
x—5    x—9    x—S    X— 6 

x-{-l     x+4:  _x-\-2    x-\-S 
x—1     x—4:    x  —  2    x  —  S 

x-\-5_x  —  Q_x  —  4_x  —  15 

x+4:    X  —  1     x  —  b     a;— 16 

o?-\-x    a'^  —  x_4acx-\-2a^  —  2c^ 
c^  —  x     c^-\-x  c'^  —  x^ 

x-{-3a     x-\-2a_x-{-a      x-\-2a 
x+a      x  —  a     x+3a    x-\-ba 

5a;-8     6x-44     10a;-8_a:-8 
'    x-2       x-1        x-l   ~'x^ 

5x-64  ,  rc-6    Ax-bb  ,  2x-ll 
a;— 13     x  —  1      X— 14       x  —  Qi 

9a;+4    3x+2       i_3a;+3     2a;-5 
15       3x-4~^   ^"     5        3x-4 

_    8a:-h5    4      Sx-a    4x-2    2x-2a 

10.  — X = 

9         5       2x-a       45       2x-a 

2/^    _    3        ,  .        4 


6. 


6. 


^^' 2{y-\)    '^'y-l    y-1    "  S{y-l) 

y    _     2y^     _l/-2__p__       9 
^^-  ^^     3to^        3~  3(2/-3)     ^ 

1/-2      97/-1      gi_2-|-?y      7?y+86 
^^'  2+2/     2(2/-2)       2     ^_2     2(7/-4) 


LITERAL  AND  FRACTIONAL  EQUATIONS  203 

^^    x-\-2  .  x-7    x+3    x-Q 
14.  -f 


16. 


16. 


X  X— 5    x+l  X— 4 

x-\-S  x  —  Q  _x-\-4:  x  —  5 

x-\-l  x-4:~  x-\-2  x^ 

x  —  5  x—4:    X— 10  x  —  9 


x+5    x+4    x+lO     x+9 


2c     ,  _     2x+3c  ,  3x+6c 

ic+4c  x+c        x-\-2c 


Exercise  115  —  Problems  in  Simple  Equations 

Solve  the  following  problems: 

1.  Separate  59  into  two  such  parts  that  4  times  the  smaller 
shall  exceed  twice  the  larger  by  26. 

2.  From  what  number  must  135  be  subtracted  to  get  273? 

3.  Find  the  number  to  which  if  329  be  added,  the  sum 
will  be  642. 

4.  What  number  must  be  multiplied  by  37  to  obtain  999? 

5.  A  is  3  times  as  old  as  B,  but  in  20  years  he  will  be  only 
twice  as  old.     Find  the  age  of  each. 

6.  What  number  must  one  divide  by  23  to  obtain  163? 

7.  Divide  220  so  that  the  quotient  of  one  part  divided  by 
the  other  is  4  and  the  remainder  20. 

8.  What  number  must  be  added  to  .378  to  give  .65? 

9.  A  is  53  years  old  and  B  is  33.     How  many  years  have 
elapsed  since  A  was  if  times  as  old  as  B? 

10.  What  number  must  one  subtract  from  3f  to  get  2 J? 

11.  Divide  $15  into  two  parts  so  that  there  are  twice  as 
many  dimes  in  the  first  part  as  there  are  5-cent  pieces  in  the 
second  part. 


204  ELEMENTARY  ALGEBRA 

12.  By  what  number  must  one  multiply  3^  to  obtain  7y? 

13.  The  difference  between  two  numbers  is  17;  and  if  4  is 
added  to  the  larger  number,  the  sum  is  4  times  the  smaller 
number.     Find  the  numbers. 

14.  Divide  $9000  into  two  parts  such  that  the  interest  on 
the  greater  part  for  2  years  at  6%  shall  be  equal  to  the  interest 
on  the  other  part  for  3  years  at  5%. 

15.  What  number  subtracted  from  164  gives  the  same 
result  as  92  added  to  the  number? 

16.  Of  what  number  is  5  J  the  three-tenths  part? 

17.  The  difference  between  two  numbers  is  32;  and  if  the 
greater  is  divided  by  the  less,  the  quotient  is  5  and  the  re- 
mainder 4.     Find  the  numbers. 

18.  By  what  number  must  one  divide  3  J  to  get  5 J? 

19.  Three  men  earned  a  certain  sum  of  money.  A  and  B 
earned  $180;  A  and  C  earned  $190;  and  B  and  C  earned  $200. 
How  much  did  they  all  earn? 

20.  What  number  is  as  much  under  7^  as  it  is  over  5 J? 

21.  The  length  of  a  rectangle  is  if  times  its  width.  If 
each  dimension  were  3  inches  less,  the  area  would  be  dimin- 
ished 279  square  inches.     Find  the  length. 

22.  What  number  lies  midway  between  3j  and  7^ ? 

23.  A  man  bought  a  coat  for  $36,  paying  for  it  in  2-dollar 
bills  and  50-cent  pieces,  giving  twice  as  many  bills  as  coins. 
How  many  bills  did  he  give? 

24.  Of  what  number  does  the  double  exceed  by  9  its  J? 

26.  A  man  invested  a  certain  sum  at  5%  and  twice  as 
much  at  6%.  His  annual  income  from  both  investments 
was  $765.     How  much  did  he  invest? 


LITERAL  AND  FRACTIONAL  EQUATIONS  205 

26.  Of  what  number  is  the  9th  part  3  less  than  its  ^? 

27.  A  had  $50  more  than  B.  A  bought  land  at  $18  an 
acre  and  had  $140  left.  B  bought  10  acres  less  at  $24  an 
acre  and  had  $30  left.     How  many  acres  did  each  buy? 

28.  A  and  B  are  106  miles  apart.  They  travel  toward 
each  other,  A  at  the  rate  of  13  miles  in  3  hours,  and  B  at 
the  rate  of  9  miles  in  2  hours.  How  far  will  each  have 
traveled  when  they  meet? 

29.  A  can  do  a  piece  of  work  in  8  (3^,  7n)*  days,  and  B 
can  do  it  in  12  (4f ,  n)*  days.  In  how  many  days  can  both 
doit? 

A  can  do  |-  of  it  in  a  day; 
B  can  do  -j^  of  it  in  a  day. 
Let  X  =  the  number  of  days  in  which  both  can  do  it. 

XX  111 

--] =  lor-  =  -H 

8      12  X     8      12 

30.  A  can  do  a  piece  of  work  in  12  (8^,  a)  days,  B  can  do 
it  in  15  (12^,  h)  days,  and  C  can  do  it  in  20  (18f ,  c)  days. 
In  how  many  days  can  all  do  it  working  together? 

31.  A  and  B  can  do  a  piece  of  work  in  8  (2,  d)  days,  and 
A  can  do  it  in  20  (4f ,  a)  days.    In  how  many  days  can  B  do  it? 

.  B  can  do  ^--^q  of  it  in  a  day. 

32.  A  boy  spent  part  of  78^  and  had  left  12  times  as  much 
as  he  spent.     How  much  did  he  spend? 

33.  Find  the  number  whose  third,  fourth,  sixth,  and 
eighth  parts  are  together  5  less  than  the  number. 

34.  John's  father  gave  him  $3  yesterday.  He  spent  50^ 
today,  and  he  still  has  ^i  more  than  he  had  day  before  yester- 
day.    How  much  has  he  left? 

*The  numbers  inside  the  parenth^es  may  be  used  instead  of  those 
outside,  if  preferred. 


206  ELEMENTARY  ALGEBRA 

35.  A  can  do  a  piece  of  work  in  12  days,  which  B  can  do  in 
18  days,  and  with  C's  help  they  can  do  it  in  4  days.  In  how 
many  days  can  C  do  the  work? 

36.  If  A  can  do  half  of  a  piece  of  work  in  10  days  and  B 
can  do  the  whole  of  it  in  15  days,  in  how  many  days  can  both 
of  them  do  it  working  together? 

37.  A  speculator  bought  two  pieces  of  land  at  the  same 
price.  He  sold  one  piece  at  a  profit  of  $1700  and  the  other  at 
a  loss  of  $900,  receiving  twice  as  much  for  one  piece  as  for 
the  other.     How  much  did  each  piece  cost  him? 

38.  At  what  rate  per  annum  will  $3600  give  $270  interest 
in  one  year  8  months? 

Let  r  =  the  rate  %  per  annum. 

39.  What  sum  must  be  invested  at  5%  to  give  a  quarterly 
income  of  $105? 

40.  What  sum  put  at  interest  at  5%  per  annum  will 
amount  to  $6000  in  1  year  9  months? 

41.  A  father  is  42  years  old,  and  his  son  is  y  as  old.  If 
both  live,  in  how  many  years  will  the  son  be  f  as  old  as  his 
father? 

42.  Separate  the  number  145  into  two  parts  so  that  the 
excess  of  the  greater  over  50  shall  be  4  times  the  excess  of 
50  over  the  smaller. 

43.  If  f  of  a  certain  principal  is  invested  at  5%  and  the 
remainder  at  4%,  the  annual  income  from  both  investments 
is  $660.     Find  the  whole  sum  invested. 

44.  The  width  of  a  room  is  f  of  its  length.  If  the  length 
were  4  feet  less  and  the  width  4  feet  more,  the  room  would  be 
square.     Find  the  dimensions  of  the  room. 


LITERAL  AND   FRACTIONAL  EQUATIONS  207 

45.  A  man  invested  $20,000,  part  of  it  at  5%  and  tiie 
remainder  at  6%.  The  interest  on  the  former  for  2  years  is 
the  same  as  the  interest  on  the  latter  for  2  years  6  months. 
How  much  was  invested  at  each  rate? 

46.  A  man  invested  J  of  his  money  in  4%  bonds,  f  of  it 
in  5%  bonds,  and  the  remainder  in  6%  bonds,  buying  them 
all  at  par.  His  annual  income  from  the  whole  investment 
amounts  to  $2550.     Find  his  whole  investment. 

GENERAL   PROBLEMS 

258.  A  general  problem  is  a  problem  all  of  the  numbers  in 
which  are  general  numbers. 

It  is  therefore  evident  that  the  solution  of  a  general  prob- 
lem involves  a  literal  equation.    For  example: 

The  sum  of  two  numbers  is  w,  and  the  larger  number  is  n 
times  the  smaller.     Find  the  numbers. 

Let  X  =  the  smaller  number, 
and  nx  =  the  larger  number. 

x-{-nx  =  m 

m  mn 

Solvmg,     x== and  nx= 

l-\-n  l-\-n 

The  result  obtained  in  solving  a  general  problem  is  a  form- 
ula for  solving  all  problems  of  that  type. 

To  find  the  smaller  number,  divide  the  sum  of  the  numbers 
by  1  plus  the  ratio  of  the  two  numbers. 

To  find  the  larger  number,  divide  the  product  of  the  sum 
and  ratio  by  1  plus  the  ratio  of  the  numbers. 

These  are  the  rules  for  finding  any  two  numbers  when  their 
sum  and  their  ratio  are  known. 

259.  Generalization  in  algebra  is  the  process  of  solving 
a  general  problem  and  interpreting  the  formula  obtained  as  a 
rule  for  solving  all  problems  of  that  type.        , 


208  ELEMENTARY   ALGEBRA 

Exercise  116 

1.  The  larger  of  two  numbers  is  7  times  the  smaller,  and 
their  sum  is  1488.     Find  the  numbers. 

m  mn 

=  1488-^8  =  (1488X7)  -^8 

l-\-n  1+n 

2.  The  smaller  of  two  numbers  is  f  of  the  larger,  and 
their  sum  is  21.  '  Find  the  numbers. 

3.  If  two  numbers  are  added,  the  result  is  2769,  and  one 
is  8f  times  the  other.    Find  the  two  numbers. 

4.  The  sum  of  two  numbers  is  s,  and  the  difference  of  the 
same  numbers  is  d.     Find  the  numbers. 

Let  X  =  the  larger  number, 
and  s— a;  =  the  smaller  number. 

x  —  {s—x)=d 
Solvmg,     x  = and  s—x= 

6.  Read  these  formulas  as  rules  for  finding  two  numbers 
when  their  sum  and  their  difference  are  known. 

6.  The  sum  of  two  numbers  is  768,  and  their  difference  is 
116.     Find  the  numbers. 

s+d  '768  +  116         s-d    768-116 
2    "        2         ^"       2    ~        2 

7.  A  man  sold  a  piece  of  land  for  $6800  and  gained  the 
same  sum  he  would  have  lost,  if  he  had  sold  it  for  $5200. 
How  much  did  he  pay  for  the  -land? 

8.  The  sum  of  two  numbers  is  a,  and  m  times  the  smaller  is 
equal  to  n  times  the  larger.     Find  the  numbers. 

Let  L  =  the  larger  number, 
and  a  —  L  =  the  smaller  number. 

am—mL=nL 

Solvmg,     L  = and  a— L  = 

m-f-n  m+n 


LITERAL  AND   FRACTIONAL  EQUATIONS  209 

9.  Read  these  formulas  as  rules  for  finding  the  two  num- 
bers in  any  problem  of  this  type. 

10.  The  sum  of  two  numbers  is  472,  and  3  times  the  larger 
is  equal  to  5  times  the  smaller.     Find  the  numbers. 

am   _ 472X5  an   _ 472X3 

m+n         8  m-\-n         8 

11.  Divide  the  number  c  into  two  parts  so  that  a  times  the 
larger  part  equals  b  times  the  smaller. 

12.  A  boy  bought  oranges  at  a  cents  apiece  and  had  b 
cents  left.  At  m  cents  apiece,  he  would  have  needed  n 
cents  more  to  pay  for  them.     How  many  did  he  buy? 

13.  According  to  the  conditions  of  the  preceding  problem, 
is  m  greater  or  less  than  a?     Show  why. 

14.  Make  a  particular  problem  which  may  be  solved  by  the 
formula  obtained  in  problem  12. 

15.  The  length  of  a  rectangle  is  m  times  its  width,  and  the 
perimeter  is  n  feet.     Find  the  dimensions. 

16.  Make  a  particular  problem  which  may  be  solved  by  the 
formula  obtained  in  problem  15. 

17.  The  sum  of  two  numbers  is  s.  m  times  their  sum  equals 
n  times  their  difference.     Find  the  numbers. 

18.  To  make  the  preceding  problem  true,  is  m  greater  or 
less  than  n?     Give  your  reason. 

19.  A  rectangle  is  a  feet  longer  than  it  is  wide,  and  the 
perimeter  is  p  feet.     Find  the  dimensions. 

20.  Make  a  particular  problem  which  may  be  solved  by 
the  formula  obtained  in  problem  19. 

21.  Find  two  parts  of  a  such  that  the  quotient  of  the  greater 
divided  by  the  less  shall  be  m  divided  by  n. 

22.  A  can  do  a  piece  of  work  in  a  days,  B  in  6  days,  and  C 
in  c  days.     In  how  many  days  can  all  working  together  do  it? 


210  ELEMENTARY   ALGEBRA 

SOLUTION   OF  FORMULAS 

260.  The  student  of  physics  and  higher  mathematics  will 
often  find  it  necessary  to  solve  formulas.     For  example: 

The  distance  passed  over  by  any  body  moving  with  a 
uniform  velocity  in  any  number  of  units  of  time  is  the  product 
of  the  velocity  and  the  time. 

This  law  expressed  in  a  formula  is —    , 

d  =  vt 

Solving  this  equation  for  v  and  t,  we  have 
v  —  d-T-t  and  t  =  d-i-v 

What  is  the  average  velocity  of  a  train,  if  it  runs  448 
miles  in  16  hours? 

(^-^^  =  448-^16 

261.  The  interest  is  the  product  of  the  principal,  the  rate 
expressed  as  hundredths,  and  the  time. 

i=prt 

It  must  be  remembered  that  r  in  this  formula  represents  the 
rate  per  annum  and  t  the  number  of  years. 

Solving  this  formula,  or  literal  equation,  for  p,  r,  and  t, 
we  have  the  following  formulas : 

p  =  i-i-rt  r  =  i-^pt  t  =  i-^pr 

1.  What  sum  put  at  interest  at  6%  for  1  year  4  months 
will  yield  $60  in  interest? 

^^r^  =  60^yfo•i  =  750 

2.  At  what  rate  per  annum  will  $1300  amount  to  $1391  in 
1  year  4  months  24  days? 

z>  p^  =  91 -T- 1300  •  If^  =  .05 

3.  In  how  many  years,  months,  and  days  will  $2200 
amount  to  $2345  at  5%  per  annum? 

^>pr=143-^2200•lfo=ltf  =1  yr.  3  mo.  18  da. 


SOLUTION  OF  FORMULAS  211 

262.  The  ratio  of  the  circumference  of  any  circle  to  its 
diameter  is  approximately  3.1416.  The  exact  value  is  repre- 
sented by  TT. 

The  formulas  for  the  circumference  of  a  circle  are 
c  =  7rd  and  c  =  27rr, 
in  which  c  is  the  circumference,  d  the  diameter,  and  r  the 
radius. 

Solve  c  =  Tvd  for  d^  and  c  =  2irr  for  r,  and  read  results 
as  rules  for  finding  d  and  r. 

263.  Denoting  the  area  by  A,  the  base  by  h,  and  the  alti- 
tude by  h,  the  formulas  for  the  area  of  a  triangle  are : 

A  =  b.|  A  =  h|  A  =  bh.^ 

The  area  of  any  triangle  is  the  product  of  the  base  and  half 
the  altitude,  the  altitude  and  half  the  base,  or  half  the  product 
of  the  base  and  altitude. 

Solve  each  of  the  above  formulas  for  b  and  h,  and  read 
the  results  as  rules  for  finding  those  dimensions. 

264.  Primes  and  Subscripts.  Different  but  related  num- 
bers in  a  formula  are  often  denoted  by  the  same  letter  with 
different  primes  or  subscripts. 

Primes  are  accent  marks  written  at  the  right  and  above  the  number; 
subscripts  are  small  figures  written  at  the  right  and  below  the  number. 
For  example,  a',  a",  a'",  no,  n^,  n2,  n^. 

These  are  read  a  prime,  a  second,  a  third,  n  sub  zero,  n  sub 
one,  n  sub  two,  n  sub  three,  respectively. 

In  the  formula  for  the  area  of  a  trapezoid  we  shall  find 
the  two  parallel  bases  denoted  by  bi  and  62. 

265.  Denoting  the  area  by  A,  the  two  parallel  sides  or 
bases  by  61  and  62,  and  the  altitude  by  h,  the  formula  for  the 
area  of  any  trapezoid  is : 

A=*^^h 
2 


212  ELEMENTARY  ALGEBRA 

The  area  of  a  trapezoid  is  the  product  of  half  the  sum  of  the 
two  bases  and  the  altitude. 

Solve  the  above  formula  for  61,  62,  and  h,  and  read  the 
results  as  rules  for  finding  those  dimensions. 

Exercise  117  —  General  Formtilas 

Ir  —  a 

1.  Solve  the  formula  s  = for  a,  r,  and  I. 

r— 1 

2.  A  man  sold  a  piece  of  land  for  n  dollars  and  gained  a 
per  cent.     How  much  did  he  pay  for  it? 

3.  Solve  the  formula  s  =  — - —  for  a,  I,  and  n. 

4.  What  sum  must  be  invested  at  n%  per  annum  to 
yield  a  quarterly  income  of  a  dollars? 

5.  Solve  dxW\  =  d2W2  for  each  general  number. 

6.  By  selling  silk  at  m  cents  a  yard,  a  merchant  lost  b%. 
Find  the  cost  per  yard. 

7.  Solve  V2t  =  Vit-\-n  for  Vi,  V2,  and  t. 

8.  The  length  of  a  rectangular  field  is  m  times  its  width. 
Increasing  its  length  a  rods  and  its  width  b  rods  would 
increase  its  area  n  square  rods.     Find  the  dimensions. 

9.  What  sum  put  at  interest  at  r  per  cent  per  annum  will 
amount  to  m  dollars  in  n  years? 

10.  Solve  the  formula  -—-  =  -  for  q,  v,  and/. 

q    P    f 

11.  At  what  rate  per  annum  will  a  dollars  yield  b  dollars 
interest  in  c  years? 

12.  Solve  the  formula  F  =  ~+S2  for  C. 

5 

13.  In  how  many  years  will  the  interest  on  a  dollars  amount 
to  7n  dollars  at  r%  per  annum? 

14.  Solve  the  f ormula  -  =  T-77  for  a,  g,  h,  and  L 

Q      n-\-i 


CHAPTER  XIX 

SIMULTANEOUS   SIMPLE  EQUATIONS 

ELIMINATION   BY   COMPARISON 

266.  Elimination  by  comparison  is  accomplished  by  solving 
each  equation  for  the  same  unknown  number  and  forming 
an  equation  of  the  two  values  obtained. 

Solve  the  system: 

(Qx-5y  =  15  (1) 

{3x-h2y  =  21  (2) 

Trans- '(l)6rc-5i/  =  15  (2)  3a:+2t/  =  21 

posing:  (3)  6a;  =15+5?/       (4)         Sx  =  2l-2y 

(5)  .=  1^      (6)  x  =  21-2^ 

By  comparison   (7) 
axiom,  §  15: 

Solving  (7), 
Substituting 


Ans. 


Exercise  118 
Solve  the  following,  eliminating  by  comparison: 

Ux-8y=^12  (Sx+3y=-25 

\Qy+Sx  =  SQ  \42/+6x=-28 


1. 


3. 


4x-5y  =  27  \Sx-4y=-lS 

Sx-3y  =  24:  \4x-5y=-21 

213 


214 


ELEMENTARY  ALGEBRA 


5. 


7. 


11. 


13. 


15. 


17. 


(Sx+5y  =  2S 

\2y+4x==17 

(Sx+2y  =  27 
\2x+Sy  =  2S 

(Sx+5y  =  45 
\4x+3y  =  25 

(2x+Sy  =  5S 
\7x-2y  =  2S 

6y+6x  =  47 
Sx-Sy  =  2Q 

7y-2x  =  S4: 
7x-2y  =  lQ 

(Sy+Qx  =  50 
\9x-4y  =  24: 


6. 


8. 


10. 


12. 


14. 


16. 


18. 


(Qx-\-7y=-70 
\2y+2x=-25 

Sx-7y=-40 
ix-5y=-2S 

2y-3x=-25 
2x-\-5y=-40 

(5y-4:X=-19 

\sx+2y=-2Q 

(Qy-Qx=-Ql 

\4:X+9y=-20 

Uy-9x=-52 
\Qx+Sy=-m 


Sy-7x=-lH 
4x-7y=-S2 


267.  In  eliminating  one  unknown  number  from  a  system 
of  fractional  equations,  it  is  often  best  to  proceed  without 
clearing  the  equations  of  fractions. 


2x    5y 

^+^  =  33 
5      4 


(1) 


(2) 


Multiplying  (1)  by  3  and  (2)  by  2  and  subtracting  the 
second  result  from  the  first,  we  have 

"3""T-^^ 

From  this  equation  the  value  of  y  is  found  to  be  12.     Sub- 
stituting this  value  in  (1),  the  value  of  x  is  40. 


ELIMINATION   BY  COMPARISON 


215 


268.  Systems  of  fractional  equations  having  the  unknown 
numbers  in  the  denominators,  though  not  simple  equations, 
may  be  solved  as  such  for  some  of  their  roots. 

In  solving  such  equations,  one  of  the  unknown  numbers 
should  be  eliminated  without  clearing  of  fractions.     Thus, 


(1) 


(2) 

Multiplying  (1)  by  3  and  (2)  by  2  and  subtracting  the 
second  result  from  the  first,  we  have 

2 


X    y 

=  22 

3+L 

X     y 

=  30 

y 


=  6 


from  which  y  =  ^;  and  substituting  in  (1), 
Check  by  substituting  in  (1)  and  (2) 


Exercise  119 

Solve  the  following  and  check  the  first  six: 


1.  < 


6.  < 


\Wr 

=  25 

1        1_ 

,^   y~ 

=  15 

X    y 

=  2a 

1_1_ 

.^   y 

=  26 

X    y 

--4c 

a_b_ 
.X    y 

=  M 

2. 


V  =  27 
X    y 

§-1=36 
X    y 


4.  < 


?+3 
X  y 
5_4 
X    y 


28 


=  24 


6. 


4     2 
---  =  22 

X    y 

«-?  =  15 

y   X 


216 


ELEMENTARY  ALGEBRA 


In  the  following  systems,  first  multiply  each  equation 
through  by  the  1.  c.  m.  of  the  known  factors  in  the  denomi- 
nators : 


2 

11a; 
]__ 

Ey 

1 


lly 

1    _1 

l0i~2 

1 


5x~^Sy    ^ 

1+^  =  5^ 
'Sx^y      2 


8.  < 


10. 


7x^Uy 


Sx 


2y 
2 


=  3 


3a:     5y 

'-+-  =  6 
[Qx    by 


Exercise  120 


Solve  and  check,  eliminating  by  any  method  (see  §§  119, 


165,  and  266): 

4i/-2x=16 
5x  — 3i/  =  44 

\5x+62/  =  28 

2x-|2/  =  36 
fx+2i/  =  56 


7.  < 


9.  < 


3  5 

2x  5?/_ 

2a  26^4(^ 

a;  ?/       c 

2a  26    4  c 


11. 


a;  2/  ^ 
f5i/H-6x  =  47 
\4a:+32/  =  35 


2. 


f5x+4?/  = 
\6?/+7a;  = 

•  \9a;+32/  = 

rfx+42/= 


6. 


25 
45 

-20 

•27 

44 
24 


8.  < 


10.  < 


12. 


3x     % 
4"^2 


=  -30 


4^ 


+3a:=-48 


2x     3?/ 
,32/"^  3x 

6?/-2a;  = 


27 
31 

-34 
30 


ELIMINATION   BY  COMPARISON 


217 


13. 


15.   < 


\lx-\-Sy  =  35 

^+^  =  27 
4       5 


17.  < 


19. 


21, 


24. 


27. 


--f  =10 
ax     by 

1+^  =  24 

^ax     by 


(4:X+2y  =  a 
\4x-3y=b 

Sx  —  2y  =  a 
2x-3y  =  a 


ax-\-by  =  b 
cx  —  dy  =  d 

26. 


14. 


16.   < 


2x+6y=-29 
3y+Qx=-SQ 


=  -35 


18.  < 


7y_2x 
6      5 

i— ?-=-18 
Sx    4y 

4y     3x 


/2p+18  =  5 
\  p+  9= -5 

( 14:171— l  =  2n 
\  n  — 6m  =  0 


22. 


(t+u  =13 

\lO^+w-f27  =  107i+^ 


25. 


f3n  +  15s  =  7 

\5s+12=-r 


x{a-\-b)—y(a  —  b)=4 
x{a  —  b)-\-y{a—b)  =4 


2.4a;- 1.5?/ =12 
1.2a:H-.15i/  =  42 


r+l=  —4s 
2s-13=-5r 


29. 


28. 

a(x-y)-{-b{x+y)=2c 
a{x-\-y)  +  b{x  —  y)  =  2d 


30. 


(mx+  .sy=.in 

\  .8x-.24y  =  m2 


31. 


(2-p)4  =  3g 

(2-2?)2  =  2g-4 


f  (l+a)x+(a-2)2/  =  4a 
•  \  (3-ha)x+(a-4)y  =  (ja 


218 
33. 


ELEMENTARY   ALGEBRA 


(ay-\-hx  =  2ab 
\by+ax  =  a^-\- 


34. 


35. 


a-\-b  =  x+y 
ax-\-ar  =  by-\-b'^ 

y{a-\-n)  =  x{a—n)  -\-an 
x{a-\-n)=y{a—n)-\-^an 


36. 


38. 


iay-{-bx  =  a^-\-b^ 
\ax  —  by=  —a?  — 


X\      X2 


1__1^ 

Xi      X2 


=  25 


40.  < 


rii    ^2 

i+?-=3i 

rii    n.2 


42.   < 


.44.  < 


^+^  =  12 
2/1    2/2 

12/1     2/2 

2^+1=  ^« 

1+1  =  21 
4a;     62/ 


46.  { 


x-[-y-l 


x-y+l 
y-x^-l 
x-y-\-\ 


=  10 


10 


48.  { 


^.4x     .9?/ 


37. 


a?  —  b'^  =  ax  —  by 
{a+by  =  ax-\-by    . 

3xH-42/+6 


2a;-32/+l 
^4a;+5?/-2 


=  20 


L3a:;-3i/-8 
a+3 . n—5 


41.  < 


2  6 

71+7     a-\-9 


=  17 
29 


20 


18 


43.   < 


P+l     q+l 

|-^r-^  =  45 


45. 


47. 


tp+1     q+1 

n+S     s— 4 
12         2 


n+5     s— 4 

'_6 3_ 

x—1     y—l 
5  1 


49.   < 


a:— 1     y—l 

_y I     X 

a—d    a—c 
y         X 


b-d     b- 


=  15 
=  10 

=  39 
=  37 

=  20 
=  20 


ELIMINATION  BY  COMPARISON  219 


50.   < 


— ?-*=180 

.02a;     My 


61. 


y    ^    X    _    1 


[Mx     my 


n  —  s    n-{-s    n  —  s 
X  y  1 


n-{-s    n  —  s    n+s 


PROBLEMS  IN   SIMULTANEOUS   SIMPLE   EQUATIONS 

269.  Many  problems,  which  really  contain  two  or  more 
unknown  numbers,  are  easily  solved  by  the  use  of  a  single 
equation  containing  but  one  unknown  number. 

This  method  is  advisable  only  when  the  relations  between 
the  unknown  numbers  are  so  simple  that  all  of  them  can  be 
expressed  in  terms  of  a  single  unknown. 

In  other  problems  it  is  better  to  introduce  as  many  equa- 
tions as  there  are  unknown  numbers. 

When  using  a  system  of  two  or  more  equations  to  solve 
problems,  enough  conditions  must  be  expressed  in  the  prob- 
lem to  furnish  as  many  independent  equations  as  there  are 
unknown  numbers  to  be  found. 

Exercise  121  —  Problems  in  Two  Unknowns 

Solve  the  following  problems  using  equations  involving 
two  or  more  unknown  numbers : 

1.  The  sum  of  two  numbers  is  148,  and  their  difference  is 
38.     Find  the  numbers. 

2.  The  larger  of  two  numbers  is  3^  times  the  smaller,  and 
their  sum  is  324.     Find  the  numbers. 

3.  A  man  changed  $7  into  dimes  and  nickels,  receiving 
111  coins.     How  many  of  each  did  he  have? 

4.  Of  two  consecutive  numbers,  f  of  the  smaller  number 
exceeds  ^  of  the  larger  by  6.     Find  the  numbers. 

5.  Divide  118  into  two  parts  so  that  7  times  the  smaller 
part  shall  exceed  3  times  the  larger  by  100. 


220  ELEMENTARY  ALGEBRA 

6.  If  the  pupils  of  a  class  are  seated  3  on  each  bench, 
5  pupils  must  stand.  If  4  are  put  on  each  bench,  one  seat 
is  not  occupied.     How  many  pupils  are  in  the  class? 

7.  Half  the  sum  of  two  numbers  is  73,  and  4  times  their 
difference  is  128.     Find  the  numbers. 

8.  The  length  of  a  rectangle  exceeds  its  width  by  14,  and 
its  perimeter  is  116.     Find  the  dimensions. 

9.  Find  three  numbers  whose  sum  is  50,  the  first  being  20 
greater  and  the  second  15  greater  than  the  third. 

10.  In  the  equation  ax-{- by  =  32,  find  a  and  b,  if,  when  x=4t, 
y  =  2;  and  if,  when  x=10,  y=  —3. 

11.  There  are  4  more  spokes  in  each  front  than  in  each 
rear  wheel  of  a  wagon,  and  in  the  4  wheels  there  are  112 
spokes.     How  many  spokes  are  in  each  wheel? 

12.  The  sum  of  three  numbers  is  59.  The  second  is  8 
greater  then  the  first,  and  the  third  is  7  greater  than  the 
second.     Find  the  numbers. 

13.  If  3  carpenters  and  7  masons  together  receive  a  daily 
wage  of  $61.20  and  a  mason  receives  20  cents  a  day  more  than 
a  carpenter,  what  is  the  daily  wage  of  each? 

14.  Three  tons  of  hard  coal  and  2  tons  of  soft  coal  cost 
$32.  At  the  same  prices,  2  tons  of  hard  coal  and  6  tons  of 
soft  cost  $43.50.     Find  the  price  per  ton  of  each. 

15.  The  first  of  three  numbers  is  twice  the  third,  the  second 
is  5  less  than  the  first,  and  the  sum  of  the  three  numbers  is  55. 
Find  the  numbers. 

16.  A  man  invests  part  of  $3200  at  6%  and  the  rest  at  5%. 
If  the  annual  income  from  the  two  amounts  is  $180,  what  is 
the  amount  of  each  investment? 

17.  One  dimension  of  a  rectangle  is  8,  and  one  dimension 


ELIMINATION   BY  COMPARISON  221 

of  a  smaller  rectangle  is  6.     The  sum  of  the  areas  is  144 
and  the  difference  is  48.     Find  the  unknown  dimensions. 

18.  There  is  a  number  which  is  expressed  by  two  figures. 
The  digit  in  tens'  place  exceeds  the  digit  in  units'  place  by  4 ; 
and  if  31  is  added  to  the  number,  the  result  is  nine  times  .the 
sum  of  the  digits.     Find  the  number. 

Let  ^  =  the  tens'  digit, 
and  w  =  the  units' digit. 

19.  A  number  of  three  digits  is  equal  to  18  times  the  sum 
of  its  digits.  The  digit  in  tens'  place  is  3  times  the  digit 
in  units'  place;  and  if  99  is  added  to  the  number,  the  digits 
are  interchanged.     Find  the  number. 

Let  h  =  the  digit  in  hundreds'  place, 
and  ^  =  the  digit  in  tens'  place, 
and  u  =  the  digit  in  units'  place. 
lOOh-{-mt+u  =  lS{h+t-{-u) 
t  =  Su 
lOOh+10t-\-u+99  =  100u-\-10t+h 

20.  A  number  of  two  digits  is  9  less  than  7  times  the  sum 
of  its  digits.  If  18  is  subtracted  from  the  number,  the  digits 
are  interchanged.     What  is  the  number? 

21.  A  is  2  years  older,  and  C  is  2§  years  younger  than  B. 
The  sum  of  their  ages  is  73.     Find  the  age  of  each. 

22.  In  a  picnic  party  of  38  persons  there  were  -f  as  many 
men  as  women,  and  f  as  many  children  as  women.  How 
many  of  each  were  there  in  the  party? 

23.  A  boy  said  to  his  playmate,  ''Give  me  5  of  your 
marbles  and  then  we  will  have  the  same  number."  His 
playmate  replied,  "Give  me  10  of  yours  and  I  will  then  have 
twice  as  many  as  you."     How  many  marbles  did  each  have? 

24.  Find  three  numbers  whose  sum  is  168,  iff  of  the  first 


222  ELEMENTARY  ALGEBRA 

plus  f  of  the  second  plus  ^  of  the  thh'd  is  92;  and  if  when 
21  is  added  to  the  first,  the  sum  is  twice  the  third. 

Let  /  =  the  first  number, 

and  s  =  the  second  number, 

and  t  =  the  third  number. 

/+s+<  =  168 

2f    2s     t 
—+-+-  =  92 
3      5     2 

f+21=2t 

25.  The  average  weight  of  3  persons  is  164  lb.  The  aver- 
age weight  of  the  first  and  second  is  159  lb.,  and  of  the  second 
and  third  165  lb.     Find  the  weight  of  each. 

26.  In  a  company  of  29  persons  there  were  15  more  adults 
than  children  and  4  more  men  than  women.  How  many- 
persons  of  each  kind  were  there  in  the  company? 

27.  How  many  bushels  each  of  new  wheat  at  $1.05  a 
bushel  and  old  wheat  at  85^  a  bushel  may  be  mixed  to  make  a 
mixture  of  200  bushels  worth  90^  a  bushel? 

28.  The  numerator  of  the  larger  of  two  fractions  is  8,  and 
the  numerator  of  the  smaller  fraction  is  5.  The  sum  of  the 
fractions  is  lyf;  and  if  the  numerators  are  interchanged, 
their  sum  is  l|^.     Find  the  fractions. 

29.  The  sum  of  the  three  angles  of  a  triangle  is  180°. 
The  sum  of  twice  the  first  and  the  second  exceeds  the  third 
by  90°;  and  the  sum  of  the  first  and  twice  the  third  exceeds 
twice  the  second  by  70°.    Find  the  three  angles  of  the  triangle. 

30.  If  a  rectangle  of  paper  were  4  in.  shorter  and  3  in. 
wider,  the  area  would  be  2  sq.  in.  less  than  it  is.  If  a  strip  2 
in.  wide  is  cut  off  on  all  sides,  the  area  is  diminished  184 
sq.  in.     Find  the  dimensions. 

31.  In  the  equation  ax  —  by  =  20,  find  x  and  y  if  when  a  =  7, 
6  =  5;  and  if  when  a  =  8,  6  =  3^. 


ELIMINATION   BY  COMPARISON  223 

32.  The  sum  of  the  three  digits  of  a  number  is  15.  The 
digit  in  tens'  place  is  half  the  sum  of  the  other  two;  and  if 
198  is  subtracted  from  the  number,  the  first  and  last  digits 
are  interchanged.     Find  the  number. 

33.  A  man  has  $49  in  dollar  bills,  half-dollars,  and  quarters. 
Half  of  the  dollars  and  J  of  the  half-dollars  are  worth  $15.50; 
J  of  the  half-dollars  and  J  of  the  quarters  are  worth  $5. 
How  many  coins  has  he? 

34.  A  and  B  are  8  miles  apart.  If  they  set  out  at  the  same 
time  and  travel  in  the  same  direction,  A  will  overtake  B  in  4 
hours.  If  they  travel  toward  each  other,  they  will  meet  in 
1^  hours.     At  what  rate  does  each  travel? 

35.  A  man  bought  a  piece  of  land.  At  $5  less  per  acre,  he 
could  have  bought  40  acres  more  for  the  money;  at  $4  more 
per  acre,  he  could  have  bought  20  acres  less  for  the  money. 
Find  the  number  of  acres  bought  and  the  price  per  acre. 

36.  One  woman  paid  $2.75  for  7  lb.  of  cofifee  and  5  lb.  of 
sugar;  another  paid  $2.05  for  3  lb.  of  coffee  and  10  lb.  of 
rice;  another  paid  $1.02  for  7  lb.  of  sugar  and  6  lb.  of  rice. 
Find  the  uniform  price  of  each  per  pound. 

37.  A  harvest  hand  engaged  to  work  two  months,  July 
and  August,  for  his  board  and  $2.50  for  each  work-day.  For 
each  week-day  he  did  not  work  he  forfeited  50^  for  his  board. 
The  term  of  service  contained  8  Sundays.  At  settlement  he 
received  $123.     How  many  days  did  he  work? 

38.  The  sums  of  the  three  pairs  of  sides  of  a  triangle  are 
14, 15,  and  17.     How  long  is  each  side? 

39.  A  classroom  has  36  desks,  some  single  and  some  double. 
The  seating  capacity  of  the  room  is  42.  How  many  desks 
of  each  kind  are  there? 


224  ELEMENTARY  ALGEBRA 

40.  A  sold  35  sheep  to  B  and  25  to  C.  They  each  then 
had  the  same  number.  Before  A  made  these  sales,  he  had 
10  more  than  B  and  C  together.  How  many  did  each  have 
at  first? 

41.  A  boy  bought  some  peaches  at  the  rate  of  2  for  5^ 
and  some  others  at  3  for  5^,  paying  $6  for  all  of  them.  He 
sold  them  all  at  40^  a  dozen  and  made  a  profit  of  $4.  How 
many  did  he  buy  at  each  price? 

42.  A  has  his  money  invested  at  4%,  B  at  5%,  and  C  at 
6%.  A's  and  B's  annual  interest  together  is  $398;  B's 
and  C's  together  is  $441.50;  and  A's  and  C's  together  is 
$409.50.     How  much  money  has  each  one  invested? 

43.  The  width  of  a  rectangular  sheet  of  paper  is  6  inches 
greater  than  half  its  length.  If  a  strip  3  inches  wide  were 
cut  off  on  the  four  sides,  it  would  contain  360  square  inches. 
Find  the  dimensions  of  the  paper. 

44.  If  the  sum  of  f  of  the  first  of  three  numbers  and  f 
of  the  second  is  118,  the  sum  of  f  of  the  second  and  f  of  the 
third  is  93,  and  the  sum  of  f  of  the  third  and  f  of  the  first  is 
112,  what  are  the  numbers? 

45.  A  street  car  has  12  short  and  4  long  seats.  When  the 
seats  are  all  occupied,  56  persons  are  seated,  each  long  seat 
holding  6  more  passengers  than  each  short  one.  How  many 
passengers  does  each  kind  of  seat  accommodate? 

46.  The  sum  of  two  sides  of  a  triangle  is  58  feet,  and  the 
difference  is  14  feet.  The  perimeter  of  the  triangle  is  103 
feet.     Find  the  length  of  each  side. 

47.  In  8  months  a  sum  of  money  at  simple  interest  amounts 
to  $780.  At  the  same  rate,  in  14  months  it  amounts  to 
$802.50.     Find  the  sum  invested  and  the  rate. 


ELIMINATION   BY   COMPARISON  225 

48.  Angle  A  of  a  triangle  is  14°  less  than  angle  B,  and 
angle  B  is  10°  larger  than  angle  C.  How  many  degrees  are 
there  in  each  angle? 

49.  The  perimeter  of  a  triangle  is  80  feet.  Two  of  its 
sides  are  equal,  and  the  other  side  is  8  feet  longer  than  either 
of  the  others.     Find  each  side. 

50.  In  a  factory  where  600  men  and  women  are  employed, 
the  average  daily  wage  for  men  is  $3.25  and  for  women  $1.75. 
If  the  sum  paid  daily  for  labor  is  $1650,  how  many  men  and 
how  many  women  are  employed? 

51.  How  must  a  man  invest  $42,000,  partly  at  4|%, 
partly  at  5%,  and  partly  at  6%,  so  that  he  may  receive  an 
annual  income  of  $2200,  if  he  invests  f  as  much  at  4|%  as  he 
invests  at  the  two  higher  rates? 

52.  Of  three  fractions  the  sum  of  the  reciprocals  of  the 
first  and  second  is  3y^;  the  sum  of  the  reciprocals  of  the  first 
and  third  is  2^;  the  sum  of  the  reciprocals  of  the  second  and 
third  is  3^^.     Find  the  fractions. 

53.  A  miller  has  corn  worth  80^  a  bushel,  rye  worth  70^ 
a  bushel,  and  oats  worth  60^  a  bushel.  He  wishes  to  make  a 
mixture  of  200  bushels  of  the  three  kinds  worth  72^^  a  bushel, 
and  use  40  bushels  of  rj^e.  How  many  bushels  of  corn  and 
oats  must  he  use? 

64.  In  a  public  school  the  number  of  pupils  in  the  third  and 
fourth  grades  is  225;  in  the  third  and  fifth  grades,  200;  in 
the  fourth  and  fifth  grades,  185.  How  many  more  pupils  are 
in  the  third  grade  than  in  the  fifth? 

55.  If  a  fruit  dealer  had  bought  and  paid  for  90  lemons 
at  a  certain  price  he  would  have  had  75^  remaining.  If 
he  had  bought  150  lemons  at  the  same  price  he  would  have 
had  75<i^  less  than  enough  money  to  pay  for  them.  How 
much  money  did  he  have,  and  what  was  the  cost  of  10  lemons? 


226  ELEMENTARY   ALGEBRA 

THREE   OR   MORE   UNKNOWN   NUMBERS 

270.  To  find  the  values  of  three  unknown  numbers,  three 
independent  simultaneous  equations  are  necessary.  In 
general  there  must  be  as  many  independent  simultaneous 
equations  as  there  are  unknown  numbers  to  be  found. 

In  solving  systems  of  several  simultaneous  equations,  it  is 
best  to  eliminate  by  addition  or  subtraction.     Thus, 

[6.x+4?/-h2z  =  32  (1) 

To  sol ve  j  2a; + 32/ + 3  z  =  25  (2) 

[4a:-2i/+5z  =  22  (3) 

Multiplying  (2)  by  3,  6x4-9?/+92  =  75  (4) 

6a;H-4z/+2z  =  32  (1) 

Subtracting  (1)  from  (4),  5?/+7z  =  43  (5) 

Multiplying  (2)  by  2,  4x+62/+6z  =  50  (6) 

^x-2y^bz  =  22  (3) 

Subtracting  (3)  from  (6),  8i/+  2  -  28  .  (7) 

Multiplying  (7)  by  7,  562/+72=  196  (8) 

by+lz=  43  (5) 


Subtracting  (5)  from  (8),  *      51?/  =  153  (9) 

Dividing  (9)  by  51,  '  y  =  Z 

Substituting  the  value  of  y  in  (5)  or  in  (7),  the  value  of 
z  is  found  to  be  4.  Substituting  the  values  of  y  and  2;  in  (1), 
(2),  or  (3),  the  value  of  x  is  found  to  be  2. 

Check  by  substituting  the  calculated  values  of  x,  y,  and  z 
in  (1),  (2),  and  (3). 

(Qx-Zy-2z=15  (1) 

Solve,  \5x+2y-9z=13  (2) 

[        4x+3z  =  33  (3) 

Combine  (1)  and  (2)  and  eliminate  y.  Then  combine  the 
new  equation  found  with  (3)  and  eliminate  either  x  or  z. 


ELIMINATION   BY  COMPARISON  227 

271.  One  or  more  of  a  system  of  equations  may  not  contain 
all  the  unknown  numbers. 


'3x-\-Sy  =  SS 

(1) 

Solve: 

'Qy-Sz=15 

(2) 

6z-2x  =  S2 

(3) 

Multiplying  (1)  by  2, 

Qx-{-Qy  =  m 

(4) 

' 

-32+6i/  =  15 

(2) 

Subtracting  (2)  from  (4), 

6a:+32  =  51 

(5) 

Multiplying  (3)  by  3, 

-6a;+18z  =  96 

.   (6) 

Adding  (5)  and  (6), 

212=147 

z  =  7 
in  (2)  and  solving, 

y  =  Q 
n  (1)  and  solving, 

Substituting  the  value  of  z 

Substituting  the  value  of  y  i 

x  =  5 

Check  bj^  substituting  x  =  5,  y  —  Q,  and  2;  =  7  in  (1),  (2) 
and  (3) 

Exercise  122 

Solve  the  following  systems  of  equations: 

'x+y  =  lS  f2x+32/-4z=16 

y+z  =  19  2.  J4x-2!/-|-32;=45 

x+z  =  17  [sx-Sy-4z  =  28 

fm-n  =  13  (2x-hSy-4z=12 

3.  J  n-p  =  U  .           4.  <Sx-Sy-\-2z  =  dO 

[m  —  p  =  27  [4x-Qy-h5z  =  4:5 

4x-2z=18  ,              (3x-{-5y-\-2S  =  Sz 

Sx-2y=17  6.  <2x+4y-{-24:  =  2z 

7y-3z  =  2Q  Ux+2y-U  =  5z 


228 


ELEMENTARY  ALGEBRA 


(2x+3y=-32 
7.  \Qy-2z=-2Q 

[4x+Sz=-4S 


(Sx-15-\-2z  =  4y 
8.  ^3y+2x-15  =  Zz 

[5y-22-\-4:Z  =  5x 


(2x+4y-j-3z  =  S5  ' 
9.  \    x-^2z^Zy  =  2Z 


f3x+32;-42/  =  12 
10.  J32/-4z+4a:=10 

\Zy^2x-2z  =  \4 


Uy~Zz^2x  =  22 

11.  \zx-V^y-2z  =  22 

l5z+4a;-22/=18 


f2x+32-3i/  =  14 
12.  J32/+   z+2x=16 


f4t/+52+3x  =  38 

13.  ^32/+5x+4z  =  35 

i4a:+3z+5?/  =  35 


f3x+4i/+2z  =  29 

14.  J4?/+32+2a:  =  27 

l4a:-f-3z-h2i/  =  33 


15. 


18.   < 


21.   < 


a:— 2/=4 

x-\-y  =  \ 

x+y  =  a 

x-2  =  7 

16.  ■ 

x-z=\ 

17.  ^ 

x+z=b 

2/-2  =  3 

y—z=\ 

y-\-z=c 

1+1=9 

X   1/ 

a     b 

M=' 

i+i=8 

y    z 

19.  < 

b     c 

20.  < 

l-r* 

V=7 

f+?=4 
c    a 

J+r» 

1-1=2 

X  2/ 

a     b 

fi+i.. 

a^  y 

1-1  =  5 

1/  z 

22.  { 

^+-^=1 

a     c 

23.  < 

y    z 

1-1  =  7 

a;  2 

6  a 

i_i_i 

^Z      X 

CHAPTER  XX 
RATIO.     PROPORTION.     VARIATION 

RATIO 

272.  The  ratio  of  one  number  to  another  is  the  quotient 
of  the  first  number  divided  by  the  second. 

Thus,   the    ratio  of   6   to   8   is   written   6:8,   or   -,    or 

8 

6-^8,   and  the  ratio  of  a   to   6  is  written,  a  :  6,  or  -r,  or 

a-i-b.  The  first  forms  are  read:  '^6  to  8"  and  "a  to  6"; 
the  second  forms  are  read:  ''6  divided  by  8,"  ''6  over  8,"  or 
''six  8ths,"  and  "a  divided  by  6,"  or  "a  over  6,"  or 
'a  feths."  g 

The  fractional  forms,  -  and  -,  are   most  in  use  today, 

O  0 

though  the  old  form,  a  :  b,  is  still  used,  and  occurs  exten- 
sively in  mathematical  books. 

The  first  number  of  a  ratio  is  the  antecedent,  and  the 
second  is  the  consequent.  The  antecedent  and  consequent 
are  the  terms  of  the  ratio. 

A  common  fraction  may  always  be  regarded  as  a  ratio. 
The  numerator  is  the  antecedent  and  the  denominator  is  the 
consequent  of  the  ratio. 

The  value  of  a  ratio  is  the  quotient  expressed  in  its  lowest 
terms. 

Give  the  antecedent,  the  consequent,  and  the  value  of  each 
of  theiollowing  ratios : 

1.  2  :6  2.  6  :2  3.  10  : 4  4.  5  :  25 

5.  -  :  -  Q.  X  :  X  7.  —  8.  — 

229 


230  ELEMENTARY  ALGEBRA 

18  14 

'-1  ^''Sb  ''-T  ''i 

2  5  T 

13.  i?  14.  -7-^       15.  ^  :  1       16.  m2-n2 :  (m-nY 

2\  a(a;+3)  6 

273.  Measuring  is  Ratioing.  To  measure  any  kind  of 
magnitude  is  to  find  its  ratio  to  some  standard  unit  of  the 
kind  of  magnitude  being  measured.  Measured  magnitudes 
are  expressed  by  so-called  concrete  numbers,  such  as  6  in., 
10  ft.,  4  lb.,  20  acres,  5  days,  etc. 

The  value  of  a  ratio  of  two  such  numbers  is  calculated  by 
first  expressing  both  number's  in  a  common  unit,  and  then 
finding  the  value  of  the  ratio  of  these  equivalents.  Thus, 
the  ratio  of  12  in.  to  3  ft.  is  not  ^,  but  12  in.  to  36  in., 
or  if  ^  or  J. 

If  the  two  magnitudes  cannot  be  expressed  in  a  common 
unit,  it  is  without  meaning  to  speak  of  their  ratios. 

Exercise  123 
Simplify  the  following  ratios: 
1.  18  :  12  2.  24i/  :  Sy  3.  2^  :  if 

4.  1  mi.  :  660  ft.  5.  7  da.  :  7  hr.  6.  16  lb.  :  4  oz. 

7.  1001b.  :1  ton  s.  ^^^±^  -    ^^^^^^ 

x-\-y 

x-y  {a-\-by 

,^      a^-b'  _        a'+b' 

13.— r-; — j — r~7ii  14. 


a^+ab-\-b^  '  a^-ab+b^ 

n/  _         x  —  b 


16. r  17. 


a. 

a 

12. 

a' 

-V 

{a- 

-hy 

15. 

x» 

-f 

(X- 

-vY 

1  Q 

a^+¥ 

1\  '  x^-3a;-10  a^'-b' 


n 


RATIO  231 

274.  A  ratio  of  greater  inequality  is  a  ratio  in  which  the 
antecedent  is  greater  than  the  consequent.  Thus,  7  :  5  is  a 
ratio  of  greater  inequality. 

275.  A  ratio  of  less  inequality  is  a  ratio  in  which  the  ante- 
cedent is  less  than  the  consequent.  Thus,  5  :  7  is  a  ratio  of 
less  inequality. 

276.  Theorem.  A  ratio  of  greater  inequality  is  diminished, 
and  a  ratio  of  less  inequality  is  increased,  by  adding  the  same 
positive  number  to  both  its  terms. 

Illustration,     (a)  For  a  ratio  of  greater  inequality. 

Let  the  ratio  be  5:3,  or  ^,  and  the  positive  number,  n, 

b-\-n 
be  added  to  both  terms  giving  the  ratio  5+n  :  3+n,  or  -— — . 

6-\-n 

,.  Then  dividing:  3+w)  5+  n  (f 

5+|n 

-In 

Or,  since  dividend  =  divisor  X  quotient -^remainder, 

5+w     K  2n  5+n   .    ,       ^,         ^ 

TT-. =?  — TTTTT-; 7j    01"    TTI IS  ICSS  thaU     ^. 

3+n     ^     3(3+n)'        3+n  ^ 

(b)  An  illustration  for  a  ratio  of  less  inequality  is  left  to  the 
student. 

Exercise  124  —  Problems 

1.  Divide  a  scanthng  16  ft.  long  into  two  parts  that  are 
to  each  other  as  3  :  5. 

Call  one  part  Sx  and  the.  other  5x.  Note  that  Sx-\-5x  =  lQ,  and 
find  X,  Sx,  and  5x. 

2.  Divide  the  number  80  into  two  parts  that  are  to  each 
other  as  2  :  3. 

3.  Divide  an  18-foot  scanthng  into  two  parts  that  are 
as  3  :  5. 


232  ELEMENTARY  ALGEBRA 

4.  Separate  121  into  two  parts  that  are  to  each  other 
as  3  : 8. 

6.  The  value  of  a  fraction  is  f .  If  both  terms  are  increased 
by  2,  the  value  of  the  resulting  fraction  is  y.  Find  the  orig- 
inal fraction. 

Notice  that  the  numerator  of  the  original  fraction  is  to  the  denomi- 
tor  as  2  :  3. 

6.  The  numerator  is  to  the  denominator  of  a  fraction  as 
3:4.  If  the  numerator  is  increased  and  the  denominator 
diminished  by  5,  the  value  of  the  resulting  fraction  is  ^. 
Find  the  original  fraction. 

7.  The  value  of  a  fraction  is  -^.  If  4  is  subtracted  from 
both  terms  the  resulting  fraction  has  the  value  f .  Find  the 
original  fraction. 

8.  The  ratio  of  the  areas  of  two  fields  is  f .  The  larger 
field  is  25  acres.     Find  the  area  of  the  smaller  field. 

9.  The  areas  of  fields  of  the  same  shape  are  as  the  squares 
of  their  corresponding  sides.  How  do  the  areas  of  two  fields 
compare  if  a  pair  of  corresponding  sides  are  as  7  :  13? 

PROPORTION 

277.  A  proportion  is  an  equation  of  ratios. 
Examples:  §=§  and  5x  :  4x  =  5  :  4.  , 

Four  numbers,  as  a,  b,  c,  and  d,  are  said  to  be  in  proportion, 

a     c 
or  proportional,  ii  a  •.b  =  c  :  d,  or  Y  =  -r 

0     a 

278.  The  terms  of  the  ratios  are  called  terms  of  the  pro- 
portion. 

279.  Extremes  and  Means.  The  first  and  fourth  terms 
of  the  proportion  are  the  extremes  and  the  second  and  third 
terms  are  the  means. 


PROPORTION  233 

In  the  following  proportions,  form  the  product  of  the 
means  and  the  product  of  the  extremes,  and  compare  the  two 
products: 


5     15 

2    5-2? 
■  7    31 

3    5_lf 

7  _3.5 
10"  5 

3      3.75 
■  11     13.75 

a     ax 

b~b~x 

280.  Since  a  proportion  is  an  equation,  the  principles  for  the 
solution  of  equations  apply  to  proportions.     If  then  we  have 

the  proportion :  -  =  - 

0     a 

we  may  multiply  both  sides  by  the  Led.  (bd)  and  obtain: 

ad=bc 

281.  Test  of  Proportionality.     The  equation,  ad  =  bc,  fur- 
nishes the  test  of  proportionality,  which  is: 

In  any  proportion  the  product  of  the  means  equals  the  product 
of  the  extremes. 

Exercise  126 

Test  whether  the  following  expressions  are  proportions : 

o      7.__1_3_  q       8  _  32 


1. 

4. 

a 
9" 

am -{-an 

9m -{-9n 

3  ""  12 

e    i5_60  R    m_am 

n     an 


1  1  ~41 


Exercise  126 

Solve  the  following  proportions  for  x: 

^•^~¥  ^'^-iO  ^'x~^ 

-7    X  c^2  c       '^        7 


234  ELEMENTARY   ALGEBRA 

'■36-3  '•"2r~3  ■  j^"T 

10.  -^-  =  1 

x+a     3a 

282.  Mean  Proportional.  If  the  second  and  third  terms 
of  a  proportion  are  the  same  number,  as  in  5  :  15  =  15  :  45, 
this  number,  the  15,  is  a  mean  proportional  between  the 
extremes. 

Thus,  in  a  :x  =  x  :  b,  x  is  a  mean  proportional  between 
a  and  b,  and  we  have : 

x^  =  ab,  whence  x=  \/a6>  oi*,  in  words: 

A  mean  proportional  between  two  numbers  is  the  square  root 
of  their  product. 

283.  A  Third  Proportional.  In  the  proportion  a  :b  =  b  :  c, 
the  number  c  is  a  third  proportional  to  a  and  b. 

Thus  in  ^  =  f  ^,  80  is  a  third  proportional  to  5  and  20. 

284.  A  Fourth  Proportional.     A  fourth  proportional  to  the 

three  numbers,  a,  b,  and  c,  is  the  number  d  in  the  proportion 

a     c 

-  =  -.     It  is  the  number,  which  with  the  three  given  numbers, 

completes  a  four-termed  proportion. 

Thus  in  i^  =  f^,  39  is  the  fourth  proportional  to  7,  13,  and  21.  , 

«  Exercise  127 

Find  mean  proportionals  between : 

1.  3  and  27  2.  4  and  16 

3.  1  and  81  4.  a  and  b 

6.  J  and  ^^3-  6.  1  and  x^ 
4            9,Pi 

7.  Z  and  —^  6.  a+b  and  dia+by 
a         ax^ 


PROPORTION 

Find  third  proportionals  to  the  following: 

9.  2  and  6  10.  4  and  9 

11.  2  and  22  12.  J  and  2 

13.  f  and  —8  lA.  x  —  y  and  x+y 

16.  x-\-y  and  x^  —  y"^ 


235 


16.  -  and  - 

a         X 


Find  fourth  proportionals  to  the  following : 


17.  4,  8,  and  12 
19.  5,  6,  and  12^ 
21.  m,  n,  and  p 
23.  m-\-n,  m  —  n,  and  w?  —  n'^ 

1 


18.  12,  3,  and  1 

20.  a,  X,  and  ?/ 

22.  a,  a^,  and  a'' 

24.  x^  x^,  and  a;^ 


25.  a:+l, 


,  and  x—1 


x-l 

PRINCIPLES   OF   PROPORTION 

285.  Since  each  of  the  following  products  is  24,  we  may 
write  2- 12  =  3-8. 

Using  only  the  four  numbers  of  these  two  products,  we 
may  write  the  two  columns  below,  the  first  being  proportions 
and  the  second,  not  proportions. 

Test  by  §  281  the  expressions  of  both  columns  and  show 
that  the  expressions  of  the  first  column  meet  the  test,  while 
those  in  the  second  column  do  not. 


PROPORTIONS 

EXPRESSIONS  NOT  PROPORTIONS 

1.  2  :3  =  8  :12 

1.  2  :12  =  3  :8 

2.  2  :8=12  :3 

2.  2  :8=12  :3 

3.  12  :3  =  8  :2 

3.  12  :2  =  3  :8 

4.  12  :8  =  3  :2 

4.  12  :8  =  2  :3 

5.  3  :12  =  2  :8 

5.  3  :8=12  :2 

6. 

3 

2  =  8  :12 

7. 

8 

3  =  12  :2 

8. 

8 

12  =  3  :2 

236  ELEMENTARY  ALGEBRA 

6.  3  :2  =  12  :8 

7.  8  :12  =  2  :3 

8.  8  :2  =  12  :3 

Notice  that  in  the  first  column  the  proportions  are  made  by  using 
both  factors  of  one  of  the  products  as  means,  and  both  factors  of  the  other 
product  as  extremes.  In  the  second  column  notice  that  this  plan  is  not 
observed,  and  that  the  expressions  obtained  are  not  proportions. 

286.  Principle.  If  the  product  of  two  numbers  equals  the 
product  of  two  other  numbers,  the  factors  of  either  product  may 
be  made  the  means  and  those  of  the  other  product  the  extremes 
of  a  proportion. 

Suppose  a'd=b'  c 

a     c         ,  a     b 
To  prove -r  =  -,  and  -  =  -,,  etc. 
0     a  c    a 

Proof.     Divide     both     members     of     a'd  =  b'C    by    bd, 

a     c 
and  simplify,  obtaining  -r=-. 

b     a 

Also  divide  both  sides  of  a'd=b'  c  by   cd,  and  obtain 

-  =  -,  etc.     Other  proportions  are  proved  similarly, 
c     a 

See  how  many  of  the  8  possible  proportions  you  can  write 
from  the  equation  a'd=b'  c 

You  should  be  able  to  write  two,  beginning  with  any  one  of 
the  4  letters. 

Exercise  128 

Write  all  the  proportions  you  can  from  the  following 
equations : 

1.  3-12  =  4-9  3.  3-7  =  2M 

2.  2-25  =  5-10  4.  m-q^U'p 

287.  Just  as  equations  may  be  derived  from  other  equa- 
tions so  may  proportions  be  derived  from  other  proportions. 
The  principles  for  deriving  proportions  from  proportions 
are  now  to  be  established. 


PROPORTION  237 

288.  Proportion  by  Alternation.     //  four  numbers  are  in 

proportion,  they  will  be  in  proportion  by  alternation,  or  the 

means  of  the  proportion  may  be  interchanged. 

_^,    ,  .     .-  a     c  ,,       a     b 

That  IS,  if  T  =  -i,  then  -=- 

b    a  c    a 

a     c 

From  T=-iJ  we  have,  by  §  281, 

0    a 

ad  =  bc 
From  which  by  §  284  we  obtain : 
a  _b 
c    d' 
This  expresses  the  principle  that  if  the  means  of  a  pro- 
portion be  interchanged,  the  result  will  be  a  proportion. 

Deriving  a  proportion  in  this  way  is  said  to  be  taking  the 
given  proportion  by  alternation. 

289.  Proportion  by  Inversion.  //  four  numbers  are  in 
proportion  they  will  be  in  proportion  by  inversion  or  the 
two  ratios  may  be  inverted. 

That  is,  if  -  =  -,  then-  =  -. 

0    a  a     c 


From  the  proportion, 


a_  c 
b~d 
By  §  281,  we  obtain:  b-  c  =  a'd. 

From  which  by  §  284,  we  have  -=-. 

a     c 

That  is,  the  two  ratios  of  a  proportion  may  be  inverted 
without  destroying  proportionality. 

Deriving  a  proportion  in  this  way  is  called  taking  the  given 
proportion  by  inversion. 

290.  Proportion  by  Addition.  If  four  numbers  are  in 
proportion,  they  will  be  in  proportion  by  addition,  that  is 
the  sums  of  the  two  terms  of  the  ratios  will  form  a  proportion 
with  either  the  antecedents  or  the  consequents. 


238  ELEMENTARY  ALGEBRA 

0    a  a  c  0  a 

To  prove  (1)  and  (2),  proceed  by  analysis,  thus: 

ANALYSIS 

Assume  (1) = ,  or  (2)  -——  =  —1-. 

a  c  0  a 

Reduce  the  improper  fractions  to  mixed  numbers  thus : 

1+-=1+-     or     -+1=3+1 
a  c  b  a 

-ni7L  b     d  a     c 

Whence,  -  =  -  or     7-  =  - 

a     c  b    d 

PROOF 

We  may  now  construct  the  proof,  by  reversing  the  steps 
just  given. 

We  know  that    -  =  -  if  ?  =  -^.    Why? 
a     c       b     d 

Add  1  to  both  sides  of  the  equations : 

a  c  b  d 

Reducing  to  improper  fractions  we  have: 

a+6     c-\-d       ,     a+6     c-\-d 

= and     — r—  =  — —. 

a  c  b  d 

When  either  of  the  last  two  proportions  is  inferred  directly 

a     c  a     c 

from  Y  =  -;,  the  proportion,  7  =  -;,  is  said  to  be  taken  by  addi- 
0    d  0    d 

tion. 

Proportion  by  addition  is  often  called  proportion  by  composition. 

291.  Proportion  by  Subtraction.  //  four  numbers  are  in 
proportion,  they  will  be  in  proportion  by  subtraction.  That  is, 
the  difference  of  the  terms  of  each  ratio  form  a  proportion  with 
either  the  antecedents  or  the  consequents  of  the  ratios. 


PROPORTION  239 

T-  a     c     ^,        a—h     c  —  d  a  —  b     c  —  d 

If-  =  -,    then  = ,    or    — —  =  _— 

0    d  a  c  .0  d 

Use  the  method  of  analysis  just  as  it  was  used  above. 
Proportion  by  subtraction  is  often  called  proportion  by  division. 

292.  Proportion  by  Addition  and  Subtraction.  If  four 
numbers  are  in  proportion  they  will  be  in  proportion  by  addi- 
tion and  subtraction. 

^p  a     c     ^,        a+6     c-\-d 

If  -  =  -,    then  r  = -. 

0    a  a—b     c—d 

Combine  the  results  of  the  principles  of  §§  290  and  291. 

Proportion  by  addition  and  subtraction  is  often  called  proportion  by 

composition  and  division. 

Exercise  129 

1.  From  each  of  the  following,  write  a  proportion  (1)  by 
alternation,  (2)  by  inversion,  (3)  by  addition,  (4)  by  sub- 
traction, and  (5)  by  addition  and  subtraction: 

1     3_9  o    2_i4  o         6_-42  8_m+n 

1-2-6      ^.9-63      ^.  zrn     77    ^-3";^^^ 

2.  From  each  of  the  following  equations  write  a  proportion, 
commencing  with  each  of  the  four  factors;  then  take  each 
proportion  (1)  by  alternation,  (2)  by  inversion,  and  (3)  by 
addition  and  subtraction: 

1.  2-9  =  3-6  2.  3-8  =  2-12 

3.  f  •7  =  3-i-         .  4.  2.7-3  =  9-0.9 

3.  Find  values  of  x  in  the  following  proportions: 

a;-7_6  a:+5_5 

•  7       4  *  x-Z     1 

x^^]_  3a;+5_ll 

•  x+2     11  *  5a;-5     5 


240  ELEMENTARY  ALGEBRA 

*  x-3     8      .  '  a:-6        5 

4.  Divide  91  into  two  parts  that  are  to  each  other  as 

5.  Divide  m  into  two  parts  that  are  to  each  other  as  a  :  b. 

6.  The  difference  between  two  numbers  that  are  to  each 
other  as  a  :  6,  is  d.     Find  them. 

7.  What  number  must  be  added  to  each  term  of  3  :  6  =  4  :  8 
to  give  another  proportion? 

8.  By  what  number  must  each  factor  of  the  products 
25-51  and  31  -40  be  reduced  that  the  products  may  be  equal? 

9.  By  what  number  must  each  factor  of  the  product 
30-147  be  reduced  and  each  factor  of  14-62  be  increased, 
to  make  the  products  equal? 

10.  What  number  must  be  added  to  both  m  and  n  to  give 
sums  which  are  to  each  other  as  a  :  6? 

11.  What  number  added  to  m  and  subtracted  from  n 
gives  numbers  to  each  other  as  a  :  6? 

12.  The  value  of  a  fraction  is  f .  Increasing  numerator 
and  denominator  by  2  gives  a  fraction  whose  value  is  f . 
What  is  the  fraction? 

13.  The  denominator  of  a  fraction  is  6  greater  than  the 
numerator.  Reducing  both  terms  by  1  gives  a  fraction 
whose  value  is  J.     Find  the  fraction. 

14.  If  the  denominator  of  a  fraction  whose  value  is  f ,  is 
increased  and  the  numerator  decreased  by  3,  the  value  of  the 
resulting  fraction  is  ^.     Find  the  fraction. 

15.  By  what  number  must  both  terms  of  |-|  be  increased 
to  give  a  fraction  whose  value  is  ^? 

16.  The  value  of  a  fraction  is  f .  If  7  is  added  to  the 
numerator  and  2  to  the  denominator,  the  reciprocal  value  of 
the  original  fraction  is  obtained.     Find  the  original  fraction. 


VARIATION  241 

VARIATION 

293.  Direct  Variation.  Suppose  water  to  be  flowing 
through  a  tube  into  a  pail.  If  lo  denotes  the  weight  of  water 
in  the  pail  at  any  time  t  minutes  after  starting,  then  w  and 
t  have  different  values  at  different  times.  They  are  therefore 
called  variables.  If  the  flow  is  uniform  and  q  denotes  the 
weight  of  water  flowing  into  the  pail  in  one  unit  of  time, 
(1  min.),  w  =  qt. 

The  number  q  differs  from  w  and  t  in  that  q  is  constant  for  a 
given  flow. 

The  height,  h,  of  a  growing  tree  and  its  age  y,  the  price,  P, 
of  a  load  of  corn  and  the  price,  p,  per  bushel,  are  other 
examples  of  variables.     Give  other  examples. 

294.  If  in  a  given  discussion  or  problem  a  number  may  have 
many  different  values  it  is  a  variable  number,  or  a  variable. 
All  numbers  that  are  not  variables  are  constants. 

Thus,  if  y  varies  as  x  we  may  always  write : 
-=k,  a  constant,  or  y  =  kx. 

X 

Show  that  2/  is  a  function  of  x. 

Since  the  distance,  d,  that  a  train  runs  during  the  time,  t 
hours,  varies  as  t,  we  may  write : 

d  =  kt. 

If,  now,  the  train  runs  30  miles  an  hour.  A;  =  30,  and  we 
may  write :  d  =  30i . 

Show  that  d  is  a  function  of  t. 

295.  When  a  variable,  as  y,  is  so  related  to  another,  as  x, 
that  as  they  change,  their  ratio,  -,  remains   constant,    the 

X 

one  variable  is  said  to  vary  directly  as,  or  to  vary  as  the  other. 
In  symbols  this  is  written : 

yocx, 
and  read :  2/ varies  as  a;.  _.   , 


242  ELEMENTARY  ALGEBRA 

Exercise  130 

1.  Assume  the  amount,  w,  of  water  in  a  barrel  to  vary 
as  the  tune,  t,  since  the  in-flow  began.  Write  the  general 
law  for  the  amount  of  water  in  the  barrel  at  time,  t. 

Ans.  w  =  qt 

2.  Suppose  that  after  3  minutes  of  flow  there  are  36  qt. 
of  water  in  the  barrel.     Find  q  and  state  the  law  definitely. 

Substitute  it;  =  36,  and  t  =  3  in  the  general  law,  w=qi,  obtaining 
36  =  9*3,  or  9  =  12,  and  the  definite  form  of  the  law  is  then 

w  =  12t 

3.  After  2  minutes  of  flow  how  many  quarts  will  have 
passed  into  the  barrel? 

Substitute  t  =  2  in  w  =  12t,  obtaining  w  =  12'2  =  2A. 

4.  The  amount  of  water  in  a  cistern  is  assumed  to  vary  as 
the  square  of  the  time,  t,  since  the  in-flow  through  a  tube 
began.     Express  the  general  law  connecting  w  and  t. 

General  law :  w  =  qt^ 

6.  Suppose  that  after  5  minutes  there  are  225  qt.  in  the 
cistern.     Find  q  and  state  the  law  definitely. 

In  w  =qt'^,  put  w  =  225  and  t  =  5,  obtaining: 

225=5-25,  or  q  =  9, 
Definite  law,  w  =  9t^. 

6.  Find  the  quantity  of  water  in  the  cistern  after  3  minutes 
of  flow. 

Inw==  9P,  substitute  t  =  3,  giving 

w  =  9'9  =  Sl.     (81  qt.  in  cistern) 

7.  After  how  long  will  there  be  900  qt.  in  the  cistern? 

900  =  9- f^  or  <^  =  100,  and  t  =  10  (after  10  min.) 

8.  li  y  cc  X  and  y=lO  when  x  =  5,  what  is  the  law  con- 
necting X  and  y? 

We  have,  first,  y  =  kx. 

Making  y  =  lO  and  x  =  5,  10  =  5A: 

Therefore,      '  k  =  2 

Hence,  y  =  2x. 


VARIATION  243 

9.  When   a   spring    is    stretched  a()^II||][^^ 

by  a  force,  F,  the  amount  of  stretch,  { — s — . 

s,   varies    as   the    strength   of   the  ^  |)CfiMlMMMMl^ 

force,  F.     Express  the  general  law  The  stretch,  s,  varies  as 

of  stretch.  the  force,  F. 

How  is  this  law  shown  by  the  graduation  marks  of  an 
ordinary  spring  balance? 

10.  When  the  force  is  20  lb.,  the  stretch  is  5  inches.  Find 
k,  and  express  the  law  definitely. 

11.  How  much  would  a  force  of  32  lb.  stretch  the  spring? 

12.  How  many  pounds  of  force  would  have  to  be  exerted 
to  give  a  stretch  of  10  inches? 

13.  The  area,  A,  of  a  square  varies  as  the  square  of  a  side,  s. 
When  s  =  5,  A  =  25.  Find  /c,  and  express  the  law  connecting 
A  and  s  in  definite  form.     Have  you  met  this  law  before? 

14.  If  the  altitude  of  a  rectangle  is  constant,  the  area.  A, 
of  the  rectangle  varies  as  the  base,  x.     Write  the  general  law. 

16.  If  the  base  is  12,  the  area  is  96.  Express  the  law  in 
definite  form. 

16.  The  area.  A,  of  an  equilateral  triangle  varies  as  the 
square  of  a  side,  s.  Express  the  law  connecting  A  and  s  in 
general  form. 

17.  When  the  side  of  the  triangle  is  6  the  area  is  3  v|^. 
Find  k,  and  express  the  law  in  definite  form. 

18.  The  work,  w,  of  a  machine  varies  as  the  number  of 
hours,  h,  that  it  runs.  Write  the  general  law  of  work  for 
the  machine. 

19.  Working  3  hours,  the  machine  does  59,400  foot-tons 
of  work.     Express  the  law  of  the  machine  in  definite  form. 

20.  How  much  work  would  the  machine  do  in  1  minute, 
or  ^^0^  of  an  hour? 


CHAPTER  XXI 
POWERS.     ROOTS 

INVOLUTION 

296.  In  §§  140,  183,  185,  and  187  we  learned  how  to  raise 
monomials  to  any  power,  also  how  to  square  binomials  and 
polynomials.     Those  sections  should  be  reviewed  here. 

297.  Involution  is  the  process  of  raising  a  number  to  a 
power  whose  exponent  is  a  positive  integer. 

Involution  is  indicated  by  an  exponent,  and  the  exponent 
which  indicates  how  many  times  the  number  is  taken  as  a 
factor  is  called  the  exponent  of  the  power.     Thus, 

298.  The  base  of  a  power  in  involution  is  the  number  which 
is  raised  to  a  power. 

It  has  been  shown  that  to  multiply  any  power  of  a  base  by 
any  power  of  the  same  base,  the  exponents  are  added.    Thus, 

The  expression  of  this  law  in  general  numbers  is 

a'"Xa"  =  a*"+". 

299.  It  has  been  shown  that  to  divide  any  power  of  a  base 
by  any  lower  power  of  the  same  base,  the  exponent  of  the 
divisor  is  subtracted  from  the  exponent  of  the  dividend. 
Thus, 

244 


INVOLUTION  245 

The  expression  of  this  law  in  general  numbers  is 

At  this  point  it  is  necessary  to  prove  three  general  laws 
for  the  involution  of  monomials. 

300.  The  sign  of  continuation  is  a  series  of  dots  ...  It  is 
read  and  so  on,    (See  §  130.) 

POWER   OF  A  POWER 

301.  Let  a  represent  any  number,  m  any  positive  integral 
exponent  of  a,  and  n  any  positive  integer.  Then  {a"'y 
represents  any  power  of  any  power.  By  definition  of  a  power: 

(^gjm)n=^m.^m.^m.^m  ^    tO  71  f actOrS, 

-_  Qm+m+m+m   ...  to  n  terms, 

The  nth  power  of  the  mth  power  of  any  number  is  the  mnth 
power  of  the  number. 

The  expression  of  this  law  in  general  numbers  is 

Exercise  131 
Give  the  result  of  each  of  the  following : 
1.  (a^y  2.  (23)3  3^  (a;2)«  4.  (c")"»  5.  (x")^ 

6.  (a^y  7.  (52)5  8.  (x^y          9.  (c"')"         10.  (x'^y 

11.  What  power  of  3  is  (27)^?    What  power  of  2  is  (16)^? 
What  power  of  5  is  (125)'  ? 

12.  Express  (81)^  as  a  power  of  3;  of  9. 

13.  Express  (64)^  as  a  power  of  4;  of  2;  of  8. 

14.  Express  (9)^  as  a  power  of  81 ;  of  3;  of  27. 


246  ELEMENTARY  ALGEBRA 

POWER   OF  A  PRODUCT 

302.  Let  a  and  h  represent  any  two  numbers  and  n  any 
positive  integer.  Then  (aby  will  represent  any  power  of  the 
product  of  any  two  numbers.     By  definition  of  a  power : 

{ahY  =  ah-ah'ah'ah'ah  •  •  •  to  n  factors, 

=  {aaa  •  •  •  to w factors) (666  •  •  •  ton  factors), 
=  a"6" 

The  nth  power  of  the  product  of  two  or  more  numbers  is  the 
product  of  the  nth  powers  of  the  numbers. 

The  expression  of  this  law  in  general  numbers  is 

(ab)"  =  a"b". 
In  a  similar  manner  it  may  be  shown  that  the  law  holds  for 
the  product  of  any  number  of  factors.     Thus, 
(2a26"c)3  =  23a«63"c3  =  Sa^^^'c^ 

Exercise  132 
Write  the  power  of  each  of  the  following: 
1.  (2a2)3  2.  (22-32)2  3.  (a-6")2  4.  {Sab^cy 

6.  {Sx^y  6.  (43-5^)2  7.  (xH/y  8.  (2ac2.T)" 

POWER   OF   A  FRACTION 

303.  We  have  seen  that: 
'a\''     a  a  a 


© 


T  )  —T'T'T  '  '  '  to  n  factors, 

_a'a-a  '  •  •  to  ?i  factors 
b'h'b  •  •  •  to  n  factors 

~b- 
The  nth  power  of  a  fraction  is  the  nth  power  of  the  numerator 
divided  by  the  nth  power  of  the  denominator. 

The  expression  of  this  law  in  general  numbers  is 
a\"_a^ 
iJ  ~b^ 


INVOLU' 

noN 

247 

Exercise 

133 

Give  the  power  of  each  of  the  following: 

■•  (iJ 

2.  r-^Y 

\     xyj 

^-  W) 

4. 

(- 

"26; 

•■©■ 

•■  (-ST 

'■  (I?)' 

8. 

(- 

3?// 

'■  (iJ 

■«■  (-5)' 

■■■  (:?)■ 

12. 

(^ 

2«y 

■'■  (IJ 

"•  (-S)' 

...  (=?)- 

16. 

(- 

3 

2iy 

POWERS   OF   BINOMIALS 

304.  By  multiplication,  the  following  powers  of  a-\-b  and 
a—b  may  be  obtained : 

(a-fb)=^  =  a=^-}-3a2b+3ab2-hb3 

(a-b)3  =  a^-3a2b-|-3ab2-b3 

(a+b)^  =  a^+4a^b-f-6a2b2+4ab3+b^ 

(a  -  b)4  =  a^  -  4a^b +6a^b '  -  4ab3+b^ 

(a4-b)^  =  a^H-5a4b  +  10a"^b2+10a2b=^+5ab4+b5 

(a  -  b)^  =  a^  -  5a%  +  lOa^b^  -  lOa^b^ + Sab^  -  b^ 

305.  From  an  examination  of  these  powers,  or  expansions, 
considering  n  to  represent  the  exponent  of  the  power,  the  fol- 
lowing laws  hold  in  each  expansion : 

1.  Evei^y  term  of  the  expansion,  except  the  last,  contains  a; 

and  every  term,  except  the  first,  contains  h. 

2.  The  number  of  terms  in  the  expansion  is  n-fl;  that  is, 
it  is  1  greater  than  the  exponent  of  the  power. 


248  ELEMENTARY  ALGEBRA 

3.  //  both  terms  of  the  binomial  are  positive j  all  the  terms  of 
the  expansion  are  positive. 

4.  If  the  second  term  is  negative,  the  odd  terms  of  the  expan- 
sion are  positive,  the  even  terms  negative. 

5.  The  exponent  of  a  in  the  first  term  of  the  expansion  is  n, 
and  it  diminishes  by  1  in  each  succeeding  term. 

6.  The  exponent  of  b  in  the  second  term  of  the  expansion  is 
1,  and  it  increases  by  1  in  each  succeeding  term. 

7.  The  coefficient  of  the  first  term  of  the  expansion  is  1; 
the  coefficient  of  the  second  term  is  n ;  and  the  coefficient  of  any 
succeeding  term  is  found  by  multiplying  the  coefficient  of  the 
preceding  term  by  the  exponent  of  a  in  that  term,  and  dividing 
the  product  by  a  number  1  greater  than  the  exponent  of  b  in 
that  term. 

The  statement  of  these  laws  constitutes  what  is  called  the 
binomial  theorem.  The  theorem  is  true  of  all  the  examples 
given.  We  shall  take  it  for  granted  that  it  is  true  for  any 
positive  integral  power  of  a  binomial,  but  a  general  proof  lies 
beyond  the  scope  of  this  book. 

Students  will  find  it  helpful  to  memorize  the  coefficients 
of  the  1st,  2d,  3d,  4th,  5th,  and  6th  powers. 

306.  These  coefficients  may  be  arranged  in  a  table  forming 
what  is  known  as  Pascal's  Triangle,  as  follows: 
Coefficients  of  1st  power:  1         1 


Coefficients  of  2d    power: 

2 

1 

Coefficients  of  3d    power: 

3 

3 

1 

Coefficients  of  4th  power: 

4 

6 

4 

1 

Coefficients  of  5th  power: 

5 

10 

10 

5 

1 

Coefficients  of  6th  power: 

6 

15 

20 

15 

6 

Each  coefficient  is  the  sum  of  the  number  above  it  and  the 
number  to  the  left  of  the  latter. 

The  coefficients  of  two  terms  equally  distant  from  the 
first  and  last  terms  of  the  expansion  are  equal. 


INVOLUTION  249 

Exercise  134 

Expand  the  following  binomials  to  the  powers  indicated, 
reading  the  powers  at  sight,  if  possible: 

1.  {x+yy  2.  {a-^xY  3.  {b-cY 

4.  {a+yY  6.  (x-aY  6.  (c-bY 

7.  {a-\-yy  8.  {b-{-xY  9.  {a- cY 

10.  {b-\-xY  11,  iy-xY  12.  (a- cY 

13.  (a-w)'  14.  (6-a)'  15.  (a:-c)'^ 

16.  (n+xY  17.  (6-a:)5  18.  (a-xY 

19.  (a;-2/)^  20.  (a+x)«  21.  {b-xY 

22.  (a+x)8  23.  {a-xY  24.  (a-?/)" 

307.  When  a  or  b  is  1,  that  term  of  the  binomial  appears 
only  in  the  first  or  last  term  of  the  power.     Thus, 

(a+l)'  =  a'+5a4+10a3+10a2+5a+l 
(l-a)«=l-6a+15a2-20a3+15a4-6a-'+aV 

Exercise  136 

Give  the  following  powers : 

1.  {x+lY  2.  (l-aY  3.  {b-lY 

4.  (l+xY  6.  (a-lY  6.  {l-xY 

It  must  be  remembered  that  a  and   b  in  the  binomial 
theorem  of  §  305  represent  any  terms  whatever.    Observe: 

(2a2+4)3  =  (2a2)3-|-3(2a2)24-f  3(2a^)42+43 

=  8a«-h48«4+96a2-f64 

Exercise  136 

Give  the  expansions  of  the  following: 

1.  {b-2Y  2.  (3-x)^  3.  {a-4Y 

4.  {2-xY  5.  (a-3)^  6.  (6-2)7 

7.  (a;2+a:)^  8.  {a+a'Y  9.  (t^-t)^ 


250  •    ELEMENTARY   ALGEBRA 

EVOLUTION 

308.  A  root  of  a  number  is  one  of  the  equal  factors  whose 
product  is  the  number. 

Thus,  2  is  a  root  of  8,  16,  32,  64,  etc. 

3  is  a  root  of  9,  27,  81,  243,  etc. 

5  is  a  root  of  25,  125,  625,  etc 

Roots  are  named  from  the  number  of  equal  factors  that 
make  the  number.     See  two  definitions  §  190. 

What  root  of  16  is  2?  What  root  of  16  is  4?  What  root 
of  64  is  2?    What  root  of  64  is  4?    What  root  of  81  is  3? 

309.  Evolution  is  the  process  .of  finding  a  root,  or  one  of 
the  equal  factors,  of  a  number. 

Evolution  is  indicated  by  the  radical  sign  \/~which  is 
placed  before  the  number. 

The  radical  sign  alone  indicates  the  square  root.  If  any 
other  root  is  required,  it  is  indicated  by  a  small  figure  called 
the  index  of  the  root,  written  in  the\/  of  the  radical  sign, 
thus: 

\/l6,  v^,  \^,  V^,  V^ 

A  symbol  of  aggregation  with  the  radical  sign  indicates  the 
part  of  the  expression  that  is  affected  by  the  sign. 

Thus,  \/254-24  means  the  sum  of  \/25  and  24,  while 
\/25+24  means  the  square  root  of  the  sum  of  25  and  24. 
The  long  bar  above  is  a  vinculum.    See  §  65. 

Any  root  of  a  number  indicated  by  the  radical  sign  is 
called  a  radical. 

Since  evolution  is  the  reverse  of  involution,  the  nth  root  of  a 
is  a  number  the  nth  power  of  which  is  a. 


EVOLUTION  251 

ROOT   OF  A  POWER 

310.  Since  (a"^)"  =  a'"'S 

V  a»»«  =  a"*, 
by  extracting  the  nth  root  of  both  members, 

The  nth  root  of  a  power  is  obtained  by  dividing  the  exponent 
of  the  power  by  n. 

Exercise  137 

1.  How  would  you   find   the   square   root   of   a   power? 
The  cube  root?     The  fourth  root?     The  fifth  root? 

2.  Give  the  indicated  root  of  each  of  the  following: 

1.  \/a^  2.  \/^  3.  V  P  4.  v^  6.  S/Jc^^ 

ROOT   OF   A   PRODUCT 

311.  Since  {ab)"  =  a''b",  then 

Va^-  =  ab.     Why? 
The  nth  root  of  the  product  of  two  or  more  factors  is  the 
product  of  the  nth  root  of  the  factors. 

Exercise  138 
Find  the  indicated  root  of  each  of  the  following : 
1.  \/a2^  2.  -v/2W  3.  \/lQa^  4.  \/a^^ 

6.  \/x^'  6.  \/59a^  7.  v^81^4  g.  \/¥^i'^ 

\/l«X25X36  =  4X5X6  =  120 
9.  \/25X  49X121  10.  Vl6X  25X36X144 


11.  V27X 64X125  12.  a/8X 64X216X348 

By  the  same  principle,  any  root  of  a  number  may  be  found 
by  resolving  the  number  into  its  prime  factors.  Observe 
the  following: 

V'99225=  V34-52.7'^  =  9-o-7  =  315 


252  ELEMENTARY  ALGEBRA 

In  like  manner,  solve:  • 

13.  \/30625      14.  V86436       15.  \^2T9E2       16.  ^54872 

Observe,  also: 

V^45-60-80  =  a/(32-5)- (22.3-^)  •(24-5) 


=  V2'-33-53  =  60 
Solve  the  following: 

17.  \/l4X2lX42X63  18.  VT5a^Fx2lF?><35^V 

19.  >^36X63X72X98  20.  \/l2a'b' X bWc^ X 72a' c^ 

21.  V(^'+a:-2)(a:2-x-6)(x2-4x+3) 

• 

ROOT    OF   A   FRACTION 

312.  From  the  law,         -j—     =t — ,  we  have — 


b^ 


The  nth  root  of  a  fraction  is  the  nth  root  of  the  numerator 
divided  by  the  nth  root  of  the  denominator. 

Exercise  139 
Give  the  following  indicated  roots : 


256a^a;8 


'625xV 

313.  A  root  is  called  an  odd  root,  if  its  index  is  an  odd  num- 
ber; an  even  root,  if  its  index  is  an  even  number. 


NUMBER   OF   ROOTS 


314.  Since  8X8  =  64,  the  square  root  of  64  is  8,  and  since 

( -  8)  X  ( -  8)  =  64,  the  square  root  of  64  is  also  -  8. 


EVOLUTION  253 

It  is  evident  that  every  positive  number  has  two  square 
roots,  one  positive  and  the  other  negative. 

It  may  also  be  shown  that  every  number  has  three  cube 
roots,  four  fourth  roots,  and  so  on. 

IMAGINARY   ROOTS 

315.  The  square  root  of  —25  is  not  5,  for  5^=  +25;  neither 
is  the  square  root  —5,  for  (  —  5)^=  4-25. 

The  square  root  of  —25  is  therefore  impossible,  as  is  the 
square  root  of  any  other  negative  number. 

We  can  only  indicate  the  square  roots  of  a  negative  num- 
ber.    The  square  roots  of  —  25  may  be  written 

\/-25  and  -  \/^^ 

316.  An  imaginary  number  is  an  indicated  even  root  of  a 
negative  number. 

317.  Since  no  even  power  is  negative,  all  even  roots  of 
negative  numbers  are  imaginary. 

We  shall  learn  later  that  imaginary  numbers  are  as  real  as  any  other 
numbers,  but  the  old  name  imaginary  still  clings  to  mathematical 
literature. 

318.  The  system  of  numbers  as  presented  in  arithmetic 
consisted  of  integers  and  fractions. 

Early  in  our  study  of  algebra  the  number  system  was 
extended  to  include  negative  numbei*s. 

Now  the  number  system  is  further  extended  to  include 
imaginary  numbers.     These  will  be  studied  later. 

319.  A  real  number  is  a  number  that  does  not  involve  an 
even  root  of  a  negative  number. 

SIGNS   OF  REAL  ROOTS 

320.  Since  even  powers  are  positive,  even  roots  of  positive 
numbers  are  either  positive  or  negative. 


254  ELEMENTARY  ALGEBRA 

To  indicate  that  a  root  is  positive  or  negative,  the  double 
sign,  read  plus  or  minus,  is  generally  used: 
V^=±a2  ^^=r±a;2  ^"64=  ±8  \/8l=±3 

321.  Since  odd  powers  have  the  same  sign  as  the  number 
involved,  odd  roots  have  the  same  sign  as  the  number.     Thus, 

^ySa^  =  2a^,   \/^^^=-x',    ■</32^'  =  2a,    i/-24S¥=-Sb 

322.  The  principal  root  of  a  number  is  the  real  root  which 
has  the  same  sign  as  the  number  itself. 

The  principal  square  root  of  49  is  7,  not  —7.  The  principal  cube 
root  of  125  is  5;  of  —125  is  —5. 

TO   FIND   THE   REAL   ROOTS   OF   MONOMIALS 

323.  Rule. —  Find  the  required  root  of  the  coefficient,  and 
divide  the  exponent  of  each  letter  by  the  index  of  the  root. 

Give  odd  roots  the  sign  of  the  number  itself,  and  give  even 
roots  of  positive  numbers  the  double  sign. 

Exercise  140 
Give  the  following  roots : 
1.  v^32?«        2.  Vl^  3.  V^^^  4.  -^-x^if 

6.  ^y27a^-       6.  \^i^'  7.  a/S^^         8.  i/-a'b^^ 

9.  ^ysTx^^      10.  V-^a'c'      11-  \^^^'        12.  -yy-xHf 

SQUARE   ROOT    OF   A   POLYNOMIAL 

324.  As  we  have  learned,  §§192  and  193,  the  square 
root  of  all  trinomial  squares,  and  the  square  root  of  soint' 
polynomial  squares,  may  be  determined  by  inspection. 

We  shall  now  show  how  to  extract  the  square  root  of  any 
polynomial  square  by  the  use  of  the  following  formula. 

(a+b)2  =  a2+2ab+b2 

Since   {a+b)-  =  a-  +  2ab  +  b'^,   the   square   root  of  the   tri- . 


EVOLUTION  255 

nomial  square  is  a-^h.     Comparing  a^-{-2ab-\-b-  in  this  iden- 
tity with  its  square  root,  we  observe: 

1.  The  first  term  of  the  root  is  the  square  root  of  the  first  term 
of  the  arranged  power. 

2.  //  the  square  of  the  first  term  of  the  root  is  subtracted  from 
the  power,  the  remainder  is  2ah-\-h'^. 

The  first  term  of  the  remainder  is  the  product  of  twice  the  first 
term  of  the  root  and  the  second  term.     Therefore, 

3.  The  second  term  of  the  root  is  found  by  dividing  the  first 
term  of  the  remainder  by  2a. 

2ab+¥={2a-\-b)b 

4.  If  we  multiply  the  sum  of  2a  and  b  by  b  and  subtract 
the  result  from  2ab-\-b'^y  the  remainder  is  0. 

The  second  member  of  this  formula  represents  the  square 
of  any  binomial ;  but  since  the  terms  of  any  polynomial  may 
be  grouped  so  as  to  form  a  binomial,  a-+2ab-\-b^  may  also 
represent  the  square  of  any  polynomial. 

If  the  root  contains  three  terms,  a^  represents  the  square  of  a  binomial, 
and  2ab  represents  twice  the  product  of  a  binomial  by  a  monomial;  if 
the  root  contains  four  terms,  a^  represents  the  square  of  a  trinomial, 
and  2ab  represents  twice  the  product  of  a  trinomial  by  a  monomial. 

325.  The  following  example  illustrates  the  process  of 
extracting  the  square  root  of  a  trinomial  square. 

9a'-\-12a^x^+4x^  |  Sa^-{-2x^ 
9a' 

6a^+2xA       +12a-V 

The  first  term  of  the  root  is  3a',  the  square  root  of  9a®,  which  we  place 
at  the  right  of  the  trinomial  square. 

Subtracting  the  square  of  3a^  from  the  trinomial,  there  remains  a  part 
that  is  represented  in  the  formula  by  2ab-\-b^. 

Dividing  the  first  term  of  the  remainder  by  Qa^,  we  obtain  the 
second  term  of  the  root,  which  is  2x". 


256  ELEMENTARY   ALGEBRA 

Multiplying  6a'+2x2  (  =  2a+6)  by  2x^  (=6),  and  subtracting  the 
result  from  12aV4-4a;^,  there  is  no  remainder. 

From  the  trinomial  we  have  subtracted  the  square  of  3a^,  twice  the 
product  of  3a^  and  2x^,  the  square  of  2x2,  and  there  is  no  remainder. 
3a' +2x2  is  the  square  root  of  the  trinomial. 

In  this  work,  the  numbers  represented  by  2a  and  2a +&  are  called 
respectively  the  partial  dinisor  and  the  cotnplete  divisor. 

Check:  Calculate  {Sa^-{-2x^)^  and  compare  the  result  with  9a" 
+  l2a^x'^-\-4x\ 

326.  Wo  observe  that  in  the  extraction  of  the  square 
root  of  a  polynomial  subtraction  is  an  essential  process;  that  is, 
the  process  consists  in  the  subtraction  from  the  polynomial 
of  the  parts  of  which  the  polynomial  is  composed.  The 
first  part  subtracted  is  the  square  of  the  first  term  of  the  root, 
and  the  second  part  subtracted  is  a  product,  which  the 
remainder  is  known  to  contain. 

327.  The  same  method  applies  to  any  polynomial  whose 
root  contains  more  than  two  terms. 

If  the  root  contains  3  terms,  the  subtraction  of  the  square  of  the 
first  term  of  the  root,  which  is  a  hinomial,  is  completed  with  the  second 
subtraction.  If  the  root  contains  4  terms,  the  subtraction  of  the 
square  of  the  first  term  of  the  root,  which  is  a  trinomial,  is  completed 
with  the  third  subtraction;  and  so  on. 

The  first  partial  divisor  is  twice  a  monomial;  the  second, 
twice  a  binomial;  the  third,  twice  a  trinomial. 

25a^-40a3x4-46a2x2-24aa:3+9x^  | ^a'-Aax-^^x^ 

25a^ 
XOa^—Aax 


—4Qa^x 
-40a3x+16aV 


10a2  -  ^ax+'^x'  +30aV 

|-h30a^x^-24a3:^+9x^ 

We  find  the  first  and  second  terms  of  the  root  as  if  we  were  getting 
the  square  root  of  a  trinomial  square. 

Multiplying  the  first  term  of  the  root,   {ba^  —  Aax),  by  2,  we  have 


EVOLUTION  257 

lOa^  —  Sax  for  the  partial  divisor.     Dividing  the  remainder  by  it,  we 
have  Sx^  for  the  next  term  of  the  root. 

Annexing  Sx^  to  the  partial  divisor,  the  complete  divisor  is  lOa^  — 
Sax-\-Sx^.  Multiplying  the  complete  divisor  by  Sx^  and  subtracting 
the  product  from  S0a^x^  —  24[ax^-\-9x*,  there  is  no  remainder,  and  the 
square  root  is  5a^  —  4:ax-\-Sx^. 

328.  Rule. —  Arrange  the  terms  of  the  polynomial  with 
reference  to  the  powers  of  some  letter. 

Find  the  square  root  of  the  first  term  as  the  first  term  of  the 
root,  and  subtract  its  square  from  the  polynomial. 

Take  twice  the  root  already  found  for  a  partial  divisor,  and 
divide  the  first  terrn  of  the  remainder  by  the  partial  divisor 
for  the  second  term  of  the  root. 

Annex  the  second  term  of  the  root  to  the  partial  divisor  to 
form  a  coynplete  divisor. 

Multiply  the  complete  divisor  by  the  last  term  of  the  root 
found,  and  subtract  the  product  from  the  remainder. 

If  other  terms  remain,  proceed  as  before,  doubling  all  the 
part  of  the  root  already  found,  for  the  next  partial  divisor » 

Exercise  141 
Find  the  square  roots  of  the  following : 

1.  4:X*-\-ix^-nx^-Qx-{-9 

2.  9o*-2a2+12a3-4a+l 

3.  16a;*+4x-8a;3-loa;2-h4 

4.  4a^+13a2-6a-12a3+l 
6.  a:*+60a;+ 13x2 -10x3+36 

6.  a«+29a^-20a3-10a5+4a2 

7.  9x^+40x-r4x2-24r^+25 

8.  30a3-23a2-|-9a4-80a+64 


258  ELEMENTARY   ALGEBRA 

9.  Ax^-\-40x^-4x^-mx^-}-25x* 

10.  64a4-192a3H-64a2+120a4-25 

11.  25x«+9a:2+l-f  10a;3- 300^4 -6x 

12.  9a^-\-Qa^b-47aW-lQa¥-\-Q4b'- 

13.  x^-2x^+ox^-Hx^-i-Sx--Sx-j-4: 

14.  264a+337a2+81a^+144H-198a3 
16.  lQx*-\-7QxY-i-Q0xy^-\-4S3^ij-{-2d}/ 

16.  3664+25a4-30a35-36a63-f69ay 

17.  9x2-8x+16-10.T3-2a;^-f.T<5-h3a;4 

18.  103a2x2-f42a3a:+49a4-48ax34-04x* 

19.  4a2+2o62-fl6c2-20a6+16ac-406c 

20.  4x6H-17a;4+10.T2-12x^-4x-16r^+l 

21.  25a2-40a6+1662-f70ax4-49a:2-566x 

22.  36c4-60a2c2+25a'*-  10a2x2_|.^4_|_i2c2a;2 

4a^_4a^_lla"     6ft 
x^       x^        x'^        X 

26.  x'+—-^-^+2ax+2+^, 

a      or  x^ 

a^_2a2c_3a2     c2     3c    _9^ 
•  b*     bH     2627'rf2+2d'^16 

27.  -r+ft^x^--— +x^-5x2+— 
4  2  4 


EVOLUTION  259 

SQUARE   ROOT   OF   NUMBERS 

329.  The  squares  of  the  smallest  and  largest  numbers  of 

one,  two,  and  three  figures  are  as  follows : 

12=   1  102=   1  00  1002=   1  00  00 

92  =  81  992  =  98  01  9992  =  99  80  01 

The  number  at  the  left  of  the  sign  in  each  identity  is  the 

square  root  of  the  number  at  the  right. 

It  follows  that  if  any  square  is  separated  into  periods  of 

two  figures  each,  beginning  at  units,  the  number  of  figures  in 

the  root  is  the  same  as  the  number  of  periods. 

When  the  number  of  figures  in  the  square  is  odd,  the  left-hand  period 

is  incomplete,  containing  only  one  figure. 

If  a  represents  the  tens  and  b  the  units  in  the  square  root 
of  any  square  of  three  or  four  figures,  a +6  represents  the 
square  root,  and  a^-{-2ab-\-b'^  represents  the  square.  Then 
the  formula  expresses  this  principle : 

Any  square  of  three  or  four  figures  is  equal  to  the  square 
of  the  tens  of  its  square  root,  plus  twice  the  product  of  the  tens 
by  the  units,  plus  the  square  of  the  units. 

For  example, 

57"  =  (50+7)2  =  502+2(50X7) +7^  =  3249 
54  76170+4 
««=  49  00 

2a  =  140      5  76 
2a+6  =  144       5  76 

Separating  the  number  into  periods  of  two  figures  each,  we  find  that 
the  root  contains  two  figures,  units  and  tens. 

The  square  of  the  number  of  tens  in  the  root  is  found  wholly  in  54. 
The  largest  square  in  54  is  49,  whose  square  root  is  7.  Hence,  there  are 
not  more  than  7  tens  in  the  root. 

Since  there  are  7  tens  in  the  root,  a  =  70,  and  a^  =  4900.  Subtracting 
a^,  which  in  this  example  is  the  square  of  70,  or  4900,  from  the  number, 
we  have  a  remainder  of  576. 

This  remainder  is  the  product  of  two  factors,  represented  by  (2a-\-h)h. 
The  partial  divisor,  2a,  is  twice  70,  or  140. 


260 


ELEMENTARY   ALGEBRA 


Dividing  576  by  140,  the  quotient  is  4,  which  is  probably  the  units' 
figure  of  the  root.     The  complete  divisor,  2a+b,  is  144. 

Multiplying  144  by  4,  and  subtracting  the  product  from  576,  there  is 
no  remainder.     Hence,  70+4,  or  74  is  the  root. 

We  may  abbreviate  and  simplify  the  work  somewhat  by  omitting 
the  ciphers  and  condensing  the  other  parts,  as  follows: 


144 


54  76  74 
49 

5  76 

5  76 

87 

22  09  47 

16 

"6  09 

6  09 

188 

96  04 
81 

15  04 
15  04 

98 


At  first  we  write  only  14,  8,  and  18  of  the  partial  divisors,  and  divide 
the  remainder,  exclusive  of  the  right-hand  figure. 

If,  on  multiplying  any  complete  divisor  by  the  last  figure  of  the  root, 
the  product  is  larger  than  the  remainder,  the  last  figure  of  the  root  is 
too  large  and  must  be  diminished  by  1 . 

After  determining  the  units'  figure  of  the  root,  we  annex  it  to  the 
partial  divisor  to  form  the  complete  divisor. 


Exercise  142 

Find  the  square  root  of  the  following: 
1.  2304  2.  3481  3.  5184 

5.  4624  6.  7396  7.  5776 


4.  4761 
8.  7569 


330.  The  same  method  appUes  to  any  number  whose  root 
is  expressed  by  more  than  two  figures.  It  is  only  necessary 
to  consider  all  the  root  already  found  as  tens. 


57  15  36 1  756 
49 


44  95  70  25 1 6705 
36 


140 

8 

7 

15 
25 

1506 

90  36 
90  36 

127 

8  95 

8  89 

1340 

5 

6  70  25 
6  70  25 

When  the  partial  divisor  is  not  contained  in  the  dividend,  exclusive 
of  the  right-hand  figure,  annex  a  cipher  to  the  root  and  also  to  the  divisor, 
and  annex  the  next  period  to  the  dividend.  In  the  second  example 
above,  134  is  not  contained  in  67. 


EVOLUTION  261 

331.  Rule. —  Separate  the  number  into  periods  of  two  figures 
each,  counting  from  units. 

Find  the  greatest  square  in  the  left-hand  period  and  write 
its  square  root  for  the  first  figure  of  the  root. 

Subtract  this  square  from  the  left-hand  period,  and  to  the 
remainder  annex  the  next  period  for  a  dividend. 

Double  the  root  already  found  for  a  partial  divisor.  Divide 
the  dividend,  exclusive  of  the  right-hand  figure,  by  the  partial 
divisor,  and  annex  the  quotient  to  the  root  and  also  to  the  divisor. 
Multiply  the  complete  divisor  by  the  last  figure  of  the  root, 
subtract  the  product  from  the  dividend,  and  to  the  remainder 
annex  the  next  period  for  a  new  dividend.  Repeat  this  process, 
using  all  the  periods. 

Exercise  143 

Find  the  square  root  of  the  following  numbers: 

1.  18,769        2.  212,521        3.  3,374,567        4.  13,734,436 

5.  94,249        6.  396,900        7.  6,140,484        8.  33,860,761 

9.  57,121      10.  258,064      11.  3,717,184      12.  16,224,784 

13.  67,081      14.  544,644      16.  9,597,604      16.  76,545,001 

TO   FIND   THE   SQUARE   ROOT   OF   A  DECIMAL 

332.  Since  squaring  a  decimal  doubles  the  number  of 
decimal  places,  the  number  of  decimal  places  in  the  square 
root  of  a  decimal  is  half  the  number  of  places  in  the  number. 

Thus,  16  65  12.96  36 i  408.06 


16 

808 

65  12 
64  64 

96 

8160( 

i 

48 

36 

48 

96 

36 

262  ELEMENTARY   ALGEBRA 

333.  Rule. —  Separate  the  decimal  into  periods  of  two  figures 
each,  beginning  at  tenths. 

The  process  is  the  same  as  with  whole  numbers. 
From  the  right  of  the  root  point  off  as  many  decimal  places 
as  there  are  periods  of  decimal  places. 

Each  period  of  a  decimal  must  have  two  figures.  If  we  wish  the 
square  root  of  a  decimal  to  2  places,  we  should  have  4  decimal  places  in 
the  number;  if  we  wish  to  carry  the  work  to  3  places,  we  should  have 
6  decimal  places  in  the  number;  and  so  on.  The  number  of  decimal 
places  may  be  increased  by  annexing  ciphers. 

Exercise  144 
Find  the  approximate  square  root  of  the  following: 
^    1.  46.08  2.  .4  3.  .036  4.  5.826 

6.  315.7  6.  .8  7.  .064  8.  95.25 

TO   FIND   THE    SQUARE   ROOT   OF   A   COMMON   FRACTION 

334.  Rule. —  //  both  terms  of  a  fraction  are  squares,  find  the 
square  root  of  each  term  separately. 

If  either  term  is  not  a  square,  reduce  the  fraction  to  a  decimal, 
and  find  the  square  root  of  the  decimal. 

Exercise  145 
Find  the  square  roots  of  the  following: 
1.  4|  2.  f  3. 

6.  7f  6.  f  7. 

9.  6^  10.  f  11. 


1- 

4.  6f 

f 

8.  8| 

i 

12.  9f 

CHAPTER  XXII 
EXPONENTS.    RADICALS 

EXPONENTS 

335.  Fundamental  Laws.  Under  certain  restrictions  the 
following  laws  have  been  established: 

1.  a'"  •  a"  =  a'""*""  2.  a'"-^a'^  =  a"'~" 

3.  (a''0"  =  a'""  4.Va^=a« 

6.  (ab)"  =  a"b" 

The  restrictions  are  that  m  and  n  are  positive  integers; 
in  law  2  that  m  is  greater  than  n,  and  in  law  4  that  7n  is 
exactly  divisible  by  n. 

336.  But  m  and  ?i  might  be  0,  fractional,  or  negative  num- 
bers. The  old  definition  that  an  exponent  indicates  how 
many  times  a  number  is  taken  as  a  factor  can  have  no  meaning 
for  such  exponents.  We  now  extend  the  notion  of  exponent 
to  give  meanings  to  these  new  forms  of  exponent,  but  it  is 
convenient  to  do  this  in  such  way  that  the  five  laws  above 
shall  hold  for  the  new  forms  of  exponent. 

337.  Definition  of  aP.  In  law  2,  if  m  becomes  equal  to  n, 
we  have:  ^  „ 

—  =  a^-n  =  ao.    But,  also  —  =#  1 , 

Therefore,  a^=l,  (a  =  0). 

Any  number  {not  itself  0)  with  an  exponent  0  equals  1. 

338.  Definition  of  a^.  In  law  4,  if  m  is  not  a  multiple  of  n 
a  fractional  exponent  arises.  By  the  law  of  exponents  for 
evolution  we  have: 

a^  =  -y/a^,       a-  =  \/a,       a*  ='^/a^,  and  generally,  a«  =S/a^ 

263 


264  ELEMENTARY  ALGEBRA 

A  positive  fradionql  exponent  indicates  a  root  of  a  power  of 
the  base.  The  denominator  is  the  index  of  the  root  and  the  num- 
erator is  the  exponent  of  the  power. 

339.  Definition  of  a""".  In  law  2,  if  n  is  greater  than  m 
the  quotient  has  a  negative  exponent. 

Since  law  1  is  to  hold  for  the  new  forms  of  exponent, 
we  have :  a~"  •  a"  =  a~''+"  =  a^=l 

Therefore,  a~"-a"  =  l 

By  the  division  axiom,  §  15,  a~"  =  — 

Any  number  with  a  negative  exponent  is  equal  to  the  reciprocal 

of  the  number  with  a  numerically  equal  positive  exponent. 

2  o^  d^X'^ 

Thus,  2a-i  =  -,  2-W  =  —,  a^b-^x^y-^  =  — - 


RADICALS 

340.  A  radical  is  an  indicated  root  of  a  number.  Roots  are 
indicated  by  the  radical  sign  or  by  fractional  exponents. 

Thus,  _  • 

\/a-\-x,  5^,  (a+6)^, -y^Sa,  a%  and\/x  — 4, 

are  all  roots. 

The  radicand  is  the  number  whose  indicated  root  is  to  be 
found.  '  Thus  the  radicand  of  \/l5  is  15;  of  \/9ait  is  9a,  and 
of  -y/a—x,  it  is  a  — X. 

In  this  chapter,  except  in  §  375,  it  is  to  be  understood  that 
the  sign  \/~  means  the  positive  square  root  of  the  radicarul. 

341.  The  order,  or  degree,  of  a  radical  is  determined  by  the 
index  of  the  root. 

Thus,  \/29  is  of  the  second  order,   or   second   degree; 
V^15  is  a  radical  of  the  third  order  or  third  degree. 
What  is  the  degree  of  the  root  a*?     Of  x^? 

342.  A  rational  number  is  a  positive  or  negative  integer 
or  a  fraction  whose  terms  are  integers. 


RADICALS  265 

343.  An  algebraic  irrational  number  is  a  number  which 
cannot  be  expressed  wholly  in  rational  form.     Thus, 

-v/3,  \/9,  VT,  -s/n,  etc. 

are  irrational  numbers. 

A  surd  is  an  indicated  root  of  a  rational  number  which 
cannot  be  exactly  obtained.     Thus, 

V27,  \/25,  \/32,  V^49,  andv^64 
are  all  surds. 

If  the  radicand  of  a  surd  is  an  arithmetical  number,  the  surd 
is  an  arithmetic  surd;  if  the  radicand  is  an  algebraic  expres- 
sion, the  surd  is  an  algebraic  surd. 

An  algebraic  surd  is  not  necessarily  an  arithmetic  surd. 
For  example,  \/a  is  an  algebraic  surd,  but  it  is  not  an  arith- 
metic surd  when  a  =  1,  4,  9,  or  any  other  square. 

344.  The  coefficient  of  a  radical  is  the  rational'f actor  before 
the  radical. 

Thus,  the  (a +6),  the  2,  the  5,  and  the  (x  —  l)  are  coeffi- 
cients in 

ia+b)\/x,   2\/5,   5-v^,   and  (a;-l)v^ll. 

These  are  read:  {a-\-b)  times  the  square  root  of  x,  2  times 
the  square  root  of  5,  5  times  the  cube  root  of  9,  and  so  on. 

345.  A  surd  which  has  no  coefficient  expressed  is  called  a 
pure  surd,  or  entire  surd;  a  surd  which  has  a  coefficient 
expressed  is  called  a  mixed  surd. 

Thus,  -s/l25  and  \/17xy  are  pure  surds,  and  ab^/xy  and 
7  VTI  are  mixed  surds. 

346.  A  quadratic  surd  is  a  surd  of  the  second  order. 
The  four  examples  given  just  above  are  illustrations  of 
quadratic  surds. 

The  value  of  a  quadratic  surd,  if  an  arithmetical  number, 
may  be  obtained  correctly  to  any  number  of  decimal  places  by 
§§  333  and  334.     Surds  are  regarded  as  numbers. 


266 


ELEMENTARY   ALGEBRA 


347.  Surds  arise  in  calculating,  as  the  following  3  examples 
illustrate. 

Exercise  146 

1.  Calculate  the  diagonal  of  a  square  whose 
sides  are  1  unit  long. 

Letting  x  denote  the  length  of  the  diagonal,  we  have 
x^  =  2,  or  X  =  V  2,  which  is  a  surd. 

2.  Calculate  the  altitude  of  an  equilateral  triangle  of  side  a. 
Letting  x  denote  the  length  of  the  altitude,  we  have 

which  gives  a;=- V  3,  a  mixed  surd. 

3.  Determine  the  true  weight  of  x 
a  body  by  means  of  a  balance  of 
unequal  arms,  x  and  y. 

Let  the  true  weight  be  denoted  by  w. 
When  the  body  is  placed  in  one  pan  sup- 
pose 10  lb.  in  the  other  pan  just  balance  it. 
By  the  principle  of  the  lever: 

wx  =  \Qy 

When  the  body  is  placed  in  the  other  pan  suppose  12  lb.  just  balance 
it.     Then,  \2x  =  wy.  (2) 

^^10        ,    ,      . 
Dividing  (1)  by  (2),  we  have  —=—,  and  clearing, 

1  Ji       w 

w^  =  l20,  or  i^  =  -s/l20- 
We  may  also  find  the  ratio  of  the  unequal  arms,  by  writing  (2)  thus, 

iiry  =  l2x  (3) 

and  dividing  (1)  by  (3),  obtaining 

X    lOy 
y~12x 


wx=10y 
(1) 


Multiplying  through  by 


Extracting  square  roots. 


Z 10 


V^f ,  which  also  is  a  surd. 


RADICALS  267 

SIMPLIFICATION   OF   RADICALS 

348.  The  examples  just  given  show  the  need  for  surds  in 
calculating,  that  they  arise  just  as  other  numbers  arise  in 
problem-solving,  and  that  they  are  to  be  regarded  as  numbers. 

349.  Reduction  of  radicals  is  the  process  of  changing  their 
form  without  changing  their  value. 

Radicals  are  simplified  to  get  them  into  most  convenient 
form  for  calculating. 

A  radical  is  not  in  its  simplest  form  for  calculating: 

1.  //  the  radicand  has  a  factor  that  is  a  power  of  the  degree 
denoted  by  the  index  of  the  radical; 

2.  If  the  radicand  is  itself  a  power  of  the  degree  denoted  hy 
any  factor  of  the  index  of  the  radical; 

3.  //  there  is  a  denominator  under  the  radical  sign,  or  a 
radical  in  any  denominator. 

350.  A  radical  may  be  simplified  when  the  radicand  has  a 
factor  whose  indicated  root  can  be  found. 

By  the  law  of  §  311,  \/20=  V4^=  V?-  \/5  =  2\/5,  also 

■V^32a^  =  v^8^M[^  =  V^So^ . -^/i^  =  2a  v^ 
In  all  work  in  simplifying  surds,  only  the  principal  roots 
are  considered. 

351.  Rule. —  Take  out  of  the  radicand  the  largest  factor 
whose  indicated  root  can  he  found.  Find  this  indicated  root 
and  write  it  as  a  coefficient  of  the  other  factor. 

Exercise  147 

Simplify  the  following  surds : 

1.  \/l2              2.  \/l25  3.  v^  4.  \/l6^ 

6.  \/28               6.  V288  7.  v^  8.  \/20^^ 

9.  a/32            10.  V242  11.  v^8l  12.  ^12^^ 


268 


ELEMENTARY  ALGEBRA 


The  root  of  the  rational  factor,  when  found,  is  multiplie: 
by  the  coefficient  of  the  mixed  surd. 

Thus,  3Vl08  =  3\/36-\/3  =  18\/3. 


14.  JV24 
18.  IVIS 


13.  2V72 
17.  3V45 
21.  5\/4S 

26.  (a;+l)\/(^'-l)(^-l) 

27.  (a+3)V2a2-12a+18 


22.  f  \/63 


16.  aVaP       16.  2a^ySa^ 
19.  a;v^^       20.  Sn\/Sn* 
23.  6v^^       24.  5av/5x^ 
26.  ^y{a-{-xy{a^-x^) 
28.  v/(2/-a:)2(x2-2/2) 


352.  From  the  principle  of  §  346  many  roots  can  be  cal- 
culated approximately  from  a  few  given  values. 

For  example,  given:  \/2  =  1.414,  \/3  =  1.732,  ^2  =  1.260, 
and  ^^3  =  1.442;  to  calculate  other  roots  approximately, 

as\/l^  =  8V2  =  8-1.414  =  11.312;  and 
.^/250  =  5 v^2  =  5- 1.732  =  8.66,  etc. 


Exercise  148 

From  the  given  values  of  the  square  and  cube  roots  of  2 
and  3  calculate : 


1.  V8 

2.  V50 

3.  \/98 

4.  V162 

6.  ^ym 

6.^yu 

7.  v^l28 

8.  v^432 

9.  \/l2 

10.  \/27 

11.  \/75 

12.  V147 

13..^ 

14.  ^108 

16.  ^192 

16.  v/375 

353.  When  the  radicand  is  itself  a  power  of  the  degree 
denoted  by  a  factor  of  the  index  of  the  radical,  proceed  as 
shown  in  the  following  example: 

\/25  =    V^a/25  =  v^,     and     \/49a%*c^  =  VVT^am? 
=  \/7ab'^c^  =  bc\^7ac. 


RADICALS 


269 


354.  Rule. —  Express  the  radicand  as  a  power  and  then 
divide  the  index  of  the  root  and  the  exponent  of  the  radicand 
by  their  highest  common  factor. 


Exercise  149 

Simplify  the  following: 

i.^y25 

2.</25 

3.  v^l21 

4.  VBGa^b^ 

6.  \/36 

6.  v^49 

7.  \/125 

8.   V27a':t^ 

9.^27 

10.  \/49 

11.  \/125 

12.  \/49a:V 

13.  \/8l 

14.  \/8l 

16.  ^243 

16.  \/27a'a;^ 

355.  When  there  is  a  denominator  under  the  radical  sign 
the  form  can  be  made  more  convenient  to  calculate.  A 
radical  may  be  simpUfied  when  the  radicand  is  a  fraction, 
thus, 


^= 


24  _ 
2T- 


^ 


^3  =  !^; 


and 


/4^_    /20a6_    fir       r—r_ 
V  56  "  \  256^  "  \  25P  *  ^  ^""^  ~ 


bah 


356.  Rule. —  Multiply  both  terms  of  the  radicand  by  a  number 
that  will  make  the  denominator  a  power  whose  indicated  root 
can  be  found.     Then  proceed  as  in  §  352. 


Exercise  150 

Simplify  the 

following: 

i.v1 

2.  <^i                 3-  ^^ 

4Vjff 

6.V| 

6.^                   7.^ 

8.VS 

9.^ 

10.  \^i               11.  V^-jV 

"•^A 

270 

ELEMENTARY 

ALGEBR.\ 

. 

13.  V| 

14.  Vf 

ia.V^ 

16.  V^ 

17.  Vf 

18.  V| 

19.  VS 

20.  Vii; 

21.  V| 

22.  Vf 
W               26. 

V2  + 

23.  VS 

24.VT0 

26.  Vl- 

Vn-(i)= 

Exercise  161 
Express  with  the  radical  sign  and  simpHfy: 
1.  16'  2.  (4a)'^  3.  x'r 


6.  27^ 


6.  xV 


7.  a^b^ 


4.  2^hi' 
8.  75^:c" 


TO   REDUCE   A   MIXED   NUMBER   TO   AN   ENTIRE   SURD 

357.  A  mixed  surd  ma}^  be  reduced  to  an  entire  surd  by 
reversing  the  process  of  simplifying  surds.     Thus, 

5\/8=V'25^=V200,   and    3 ^^5  =  a^27^  =  V^l35 


Exercise  162 


Express  the  following  as  entire  surds; 


1.  2^y7  2.  f  \/27 

6.  5V3  6.  i\/48 

9.  2\/5  10.  1^32 

13.  |V|  14.  |Vp 


3.  2av5a 

7.  2x\^'Sx 

11.  5a\/do. 


16. 


4.  l|aV8a^ 

8.  l^x\/9^ 

12.  2|aV^aS2 

It 6   ./5a.T'- 


17. 


6b     IHa  Ijb     1 5ax~ 

•  2a  \  96      ^^'  lia\86r^ 

a4-2\a+2  '  \  a-f  x  a;— 4\a;- 


+1 
4 


ADDITION   AND    SUBTRACTION   OF   SURDS 

358.  Surds  are  added  or  subtracted  by  adding  or  sub- 
tracting their  coefficients. 


RADICALS  271 

359.  Similar  surds  are  surds  which  in  their  simplest  form 
are  of  the  same  degree  and  have  the  same  radicand,  such  as : 

2\/5)  4\/5>  a-\/5;  av^,  6\/.r;  and 
3-^,  5v^,  9-^,  etc. 

Two  or  more  surds  can  be  united  into  one  by  addition  or 
subtraction  only  when  they  are  similar,  as  is  shown  here: 

2V45+4\/20+5\/80-3Vl25  = 
6\/5+8V5+20-\/5  - 15\/5  =  19  V5, 


and  5^yTQ^-^yMx^-V2x^  = 

lOv^^-S v''2^-  v^2i2  =  6^2x2 

Exercise  163 

Simplify  the  following : 

1.  3V300+2V243  2.  2  ^yiSx^  -  \/M^ -\- ^y^Ox^ 

3.  4\/45+2V48-4\/27-3.\/20+2-\/l2 
4.  7-v/T75-5\/Il2  5.  4\/l6^-2\/25a3+2\/36^ 

6.  3Vn2+6\/45-3V28-f V80+3\/63 
7.  3^^375-2^/192  8.  3V24a^- \/96^+2\/54^ 

9.  2\/360-4Vl0-fV90+3'v/40-|\/250 

10.  4Vl28-3Vl62        11.  2v^81x^-3v^l6^4-v/80i^ 

12.  5>/l6-f-f  >/128-5v/54+4v^250-2aJ^'486 

TO  REDUCE  SURDS  TO  THE  SAME  ORDER 

360.  Surds  of  different  orders  are  changed  to  the  same  order 
by  expressing  the  radicals  as  fractional  exponents,  and  reduc- 
ing the  fractional  exponents  to  equivalent  fractions  having  a 
common  denominator,  and  then  expressing  the  surds  unth 
radical  signs.     Thus, 

'\/2  =  2^  andAy3  =  3^ 


272  ELEMENTARY   ALGEBRA 

The  lowest  common  denominator  of  the  exponents  is  6. 

Then,  2^  =  2"^  =  </T'  =  v^S,  and 

3^  =  35=^32=^9. 

Since  ^72=  \/8  and  v^=  \/9,  we  can  at  once  say  that 
^>  y/2. 

This  principle  enables  us  to  compare  radicals  of  different 
orders  as  to  relative  magnitude. 

The  signs  of  inequality  are  >  and  < .  The  sign,  > ,  means  greater 
than]  <  means  less  than. 

Exercise  154 
Compare  the  following  pairs  of  radicals: 

1.  \/5  and  \/7  2.  \/5  and  \/2  3.  2  V3  and  3  v^2 
4.  v^  and  v^  5.  ^  and  i/6  6.  2\/5  and  3  ^y^ 
7.  Arrange  in  order  of  value,  \/7,  \/6,  and  V^- 

MULTIPLICATION    OF   SURDS 

361.  The  product  of  two  or  more  surds  of  the  same  order 
is  found  by  law  5,  §  335. 

For  fractional  exponents  this  law  takes  the  form: 

Notice  that  this  applies  only  when  the  surds  are  of  the 
same  order.     Thus, 

V5  •  \/35  =  a/175  =  5\/7,  and 
2v^6. 3^/18  =  6^^108  =  18^ 

Exercise  166 
Multiply  as  indicated : 

1.  4\/3-3V5  2.  2\/7-3-n/7  3.  2\/5-3\/l5 

4.  5^-2v^  6.  4\/5-5V5  6.  5\/2'^^yS2 

7.  3^y4'4^yS  8.  3V6-2\/8  9.  4^yQ'2\/l2 


RADICALS 


273 


362.  To  find  the  product  of  surds  of  different  orders,  first 
reduce  them  to  the  same  order.     For  example : 

\/2- ^V^=  \/8- \/400  =  2-v/50 

When  numbers  are  large  they  can  be  better  managed  by 
the  use  of  exponents.     Thus, 

V21-   \/9S=    y/Tl'-i/T^=    x/7^3.  ^71:22=    7\/756 


Exercise  166 
Perform  the  following  multiphcations : 
1.  4V2.2>^  2.  2Vi->/| 

4.  3v^-4v^9  5.  3v^-\/| 

7.  2\/6-5\/6  8.  5V^|-V| 

12.  f Vf.f Vf.sVI 


3.  3\/2.5a/4 
6.  av^-a^v^ 
9.  2a/5-4\/2 
11.  v^-\/3a2- V^6a2^ 
13.  v^9?^- V3x-\/2^ 


14,  I  Vf -f  V|"-tVj  15.  V2a-  V^4a2-  ^Sa*-^^ 

363.  Observe  that  the  square  of  the  square  root  of  a  num- 
ber is  the  number  itself.     For  example, 

Va-\/a  =  a,  \/5-\/5  =  5,  and  VlO*  VlO  =  10 

Multiplication  of  surds  may  be  extended  to  radical  poly- 
nomials, as  shown  here: 


5+     \/3 
3+  2\/3 
15+  3\/3 
IOV3+6 

2H-13V3 

Multiply  the  following: 

3V2+2V6 
\/2-3\/6 


2-v/7-3v^ 
3\/7H-2Vx 

42-9\/7x 
4V7x-6x 

42-5-\/7x-6x 


3\/^  — Sa/?/ 


274  ELEMENTARY   ALGEBRA 

Exercise  167 
Multiply  the  following : 

1.  ^-^/ahy4+2\/a 

2.  5-f4V^by  3-3\/2 

3.  12+3\/5by  4-2\/5 

4.  3a-3 Va  by  2a+2  V« 

6.  V3-2\/6  by  V3+5\/6 

6.  2V«+3Vcby\/a+6A/c 

7.  v^-f-VS- VS  by\/2- V3H-2V5 

8.  \/S-  V5+2 V?  by  2 V3+2V5-  V^ 

9.  3\/7-2\/3+4\/5by4\/7-3\/3-\/5 

Multiply  by  inspection : 

10.  (-v/7-h2)(\/7-2)  11.  (V«+V^)(V^-fV^) 

12.  (V3+5)(V3-f 5)       IS,  {Vi-Vy)(V^-Vy) 

14.  {x-VS){x-\/^)  16.  (V8-V3)(V8-f-V3) 

16.  {^/a-hx){^/a-x)  17.  (Va;+ V^)(V^-|- V^) 

18.  (\/T0-5)(Vl0-5)       19.  (V^+\/3)(V8+\/3) 

DIVISION    OF   SURDS 

364.  The  quotient  of  two  surds  of  the  same  order  is  found 
by  the  principle  of  evolution  as  stated  in  the  formula: 

Vi^   n  /X 

Vi      Vy 


RADICALS  275 

As  in  multiplication,  so  here,  this  principle  applies  only  to 
surds  of  the  same  order.     Thus, 

2\/60-^V5  =  2\/l2  =  4-v/3, 

•^/54-^3^/3  =  |  VT8=  V2, 
also  6\/T5^2-v/T8  =  3\/f=J\/30. 


Exercise  168 

Give  these  quotients  by  inspection : 

1.  4V24-^-\/3  2.  V32 

4.  SVSI-^V^  5.  V54 

7.  3\/40^-\/2  8.  \/56 


2\/2  3.  6\/45-^2V3 

3V2  6.  2\/90^9\/5 

2\/7  9.  3V75-T-5V3 


RATIONALIZING    SURDS 

365.  Rationalizing  is  the  process  of  multiplying  a  surd 
by  a  number  that  gives  a  rational  product.  Observe  the 
following: 

V^--v/5  =  5,  ^-^  =  3, 

V8-V2  =  4,  and        ^^^•->^  =  5. 

The  rationalizing  factor  is  the  factor  by  which  a  surd  is 
multiplied  to  give  a  rational  product. 

When  the  product  of  two  surds  is  rational,  either  surd  is 
the  rationalizing  factor  of  the  other. 

Name  a  rationahzing  factor  of  each  of  the  following 
surds  and  give  the  products : 

1.  \/Q  2.  2\/l2  3.  5v^  4.  2V27a6 

5.  \/8  6.  5v^  7.  3v^  8.  4:^ymxy 

366.  A  binomial  surd  is  a  binomial  one  or  both  of  whose 
terms  are  surds.     Thus,     4+  \/5,  \/3  -  2,  and  \/6-f  \/7. 


276  ELEMENTARY   ALGEBRA 

367.  A  binomial  quadratic  surd  is  a  binomial  surd  whose 
surd,  or  surds,  are  of  the  second  order. 

368.  Conjugate  surds  are  two  binomial  quadratic  surds 
that  differ  only  in  the  sign  of  one  of  the  terms. 

For  example  a-\-  y/h  and  a—  \^b,  as  also 
\/7-V5andV7+V5 
are  conjugate  surds. 

Since  conjugate  surds  are  of  forms  a-{-b  and  a  — 6,  the 
product  of  any  two  conjugate  surds  is  rational. 

Hence  it  follows  that  any  binomial  quadratic  surd  may  be 
rationalized  by  multiplying  it  by  its  conjugate. 

Thus,  (4+\/7)(4-\/7)=9, 

and  (\/IO-V2)(\/lO+\/2)=8 

Exercise  169 

Name  a  rationalizing  factor  of  each  of  the  following  surds 
and  give  the  products : 

1.  8-Vl4               2.  2\/5-3a/2  3.  y/a^+y/a 

4.  a+2'\/6               6.  2\/T5+V7  6.  \/^-\/a-b 

7.\/7d-5               8.  \/70-3a/6  B.\/a^-\-\/a 
Exercise  160 
Rationalize  the  denominators  of  the  following: 

J          4                       2   V^-V^  3   V^+1+2 

'  3-\/5                   'VS-hV^  *\/«+l-2 

3— \/2                     ■\/a-{-\/x  a— -x/x+l 

'3+^2            "      '\/a-\G  '  a-^y/x'Ti 

a                      ^  \/a-\/x  ^  \/a^b-\-c 


a—\/b  y/x—y/a  ^Ja-^-b 


c 


RADICALS  277 

Rationalize  and  simplify  by  inspection: 

13.  -^  14.  -—  16.    -^  16.  -^-= 

17.  — ^  18.  -^  19.  ;:^— ^  20.   — ^ 

V7  V«  3v^  aV^ 

369.  Any  power  of  a  monomial  surd  is  found  by  the  rule  for 
multiplication  of  surds  of  the  same  order,  §  361. 

SQUARE   ROOT   OF  BINOMIAL   SURDS 

370.  The  square  of  a  binotnial  quadratic  surd  is  a  binomial 
quadratic  surd  one  term  of  which  is  rational.  Observe  the 
examples : 

(\/5+\/3)'  =  8+2\/l5,  and  (\/7- \/2)2  =  9-2\/T4. 

It  follows  then  that  some  binomial  surds  are  squares,  and 
the  square  root  of  them  may  be  found. 

Observe,  first,  that  the  rational  term  of  the  square  is  the 
Slim  of  the  radicands  of  the  two  given  surds;  and  second,  that 
the  irrational  term  of  the  square  is  twice  the  product  of  the 
two  given  surds. 

371.  Rule. —  Reduce  the  binomial  surd  to  the  form  a  =±=  2  -y/S- 
Separate  a  into  two  parts  whose  product  is  b.  Extract  the  square 
roots  of  the  two  parts  of  a,  and  connect  the  roots  with  the  sign  of 
the  irrational  term.     Thus, 

Vl.54-V200=  Vl5+2\/50=  VlO+V^, 
V20+4-\/24=  V20-h2V96=  \/l2-f  \/8, 
V20-5\/T2=  V20-2V75=  Vl5-  \/5. 

Exercise  161 
Find  the  square  root  of  each  of  the  following: 

1.  9+2 V20  2.  l-2-\/30  3.  13+4\/l0 

4.  5-10\/2  5.   10+4\/6  6.   10-4^12 


278  ELEMENTARY  ALGEBRA 

7.  12+4\/5  8.  15+3V6  9.  30-6\/20 

10.  2x-f-32/-2\/6x^  11.  2x+2V^'-2/' 

12.  2a  +  h-2V^Tab  13.  a2+6+2aV6 

APPROXIMATE  VALUES  OF  SURDS   " 

372.  The  approximate  value  of  a  surd  is  found  by  extract- 
ing the  indicated  root  to  the  required  degree  of  accuracy.  It 
is  frequently  necessary  to  find  the  value  of  a  fraction  with  a 
radical  denominator. 

In  such  work  much  labor  is  saved  by  first  rationalizing  the 
divisor,  or  denominator.     Thus, 

3   _3a/5     3' 2.23607 
\/5        5  5 

Simplify  each  of  the  following  divisions,  finding  the  numer- 
ical value  correct  to  5  decimal  places,  having  given  that, 

V2  =  1.41421,   \/3=  1.73205,  and  \/5  =  2.23607. 

Exercise  162 

1.  lO-^VlS  2.  13^\/27  3.  25^V20 

4.  3VT5-^2\/3         5.  18-3\/8  6.  7-^2^/75 

IRRATIONAL  EQUATIONS  IN  ONE  UNKNOWN 

373.  An  irrational,  or  radical  equation  is  an  equation  con- 
taining an  irrational  root  of  the  unknown  number.     Thus, 

V^  =  3,    V^-4  =  5,    V3aJ-5=\/^+35. 
To  solve  an  irrational  equation  the  first  step  is  to  free  the 
equation  of  radicals.     This  is  done  by  raising  both  members 
of  the  equation  to  the  same  power. 

Power  Axiom. —  The  same  powers  of  equal  numbers  are 
equal. 


RADICALS  279 

To  solve,  \/2x-6  =  3,  or  V^a;-?  =  \/2^+17 . 

Squaring,      2a;  — 5  =  9,  5a;  — 7=      2a; +17. 

The  results  of  squaring  in  these  two  examples  are  simple 
equations,  and  are  solved  as  such. 

Radical  equations  containing  more  than  one  radical  may 
have  to  be  squared  more  than  once. 

Thus,  to  solve: 

V^.T  — 5+ V''^'  =  5 

Subtracting  ■\/x,         \/.t  —  5  =  5—  \/x 

Squaring,  x  — 5  =  25  —  10  \/lc-\rX 

Uniting  terms,  10v^a;  =  30 

Dividing  by  10,  ■\/x  =  S 

Squaring,  x  =  9. 

374.  With  radical  equations  it  is  agreed  that  the  radical 
sign  shall  denote  only  principal  roots. 

Verifying,         \/9  — 5  -f  \/9  =  5 

2+3  =  5 

5  =  5 

Since  the  substitution  of  9  for  x  in  the  original  equation 
gives  an  identity,  9  satisfies  the  equation. 

Exercise  163 
Solve  and  verify  the  following: 

1.  \/^-H  =  9  2.  \^x  —  a  =  a 

3.  \/.^c+5  =  4  4.  v^x+6  =  2 

5.\/x-\-b  =  a  6.  v^a;  — 3  — 3 

7.  \^-\/x-^=V^  8.  \/^T2+\/x  =  2 

9.  \/x-7+\/7=\G  10.  V^+5-\/x=l 

11.  \/^+9-\/^-7  =  2  12.  y/x-4:+\/x  =  2 

13.  3v^-\/4a;-9=  V3  14.  \/x+ V^+8  =  4 

^^  V3^+9+5\/^^     ^  ^^  V^+9     V^T2 

15.  .  =o  16.        ,  =       , 

\/3.'r  +  9~5\/-'^-5  ;     \/^+3     \/^p-2 


280  ELEMENTARY  ALGEBRA 

19.  \/x-{-\^a+x  =  —==      20.\^a  —  \/x-{-\/x-{-a  =  \/x 
■yx^a 

21.  y/x+d+\/x-2=\/5-h4x 

375.  A  statement  may  be  in  the  forrn  of  an  irrational 
equation  which,  under  the  assumption  that  \/    shall  mean 
the  positive  square  root,  cannot  be  satisfied. 
Thus,  solving  by  the  usual  method, 
'\/x  —  5=  y/x-^^ 
we  obtain  a;  =  9.     Attempting  to  verify  we  have 

2  =  3-f5 
which  is  not  an  identity. 

Setting  aside  the  assumption  and  recalling  that  \/x  may 
be  either  the  positive  or  negative  root,  as  the  conditions  of  the 
problem  require,  and  retaining  both  signs  in  verifying,  we 
have, 

±2=  ±34-5. 

Of  these  possibihties  as  to  sign,  we  can  get  an  identity  by 
using  +2  for  \^x  —  5  and  —3  for  \/x.  It  is  worth  noting 
that  this  state  of  things  would  not  have  been  found  if  verify- 
ing had  been  omitted. 

In  squaring  a  radical  equation,  a  root  is  sometimes  i7itrO' 
duced  which  the  given   equation   did   not   contain.     Thus, 

\/4xTl=S-V^^ 
freed  of  radicals  and  solved  by  the  usual  process,  leads  to 
x  =  2  and  x  =  6 
Verifying  for  2,  3  =  3-0     This  checks. 

Verifying  for  6,  5  =  3  —  2     This  does  not  check. 

Hence,  2  satisfies  the  equation  under  the  assumption  that 
\/~indicates  only  the  positive  square  root,  while  6  does  not 


RADICALS  281 

satisfy  the  equation.     Removing  this  assumption,  however, 
and  choosing  the  -j-  or  —  sign  for  the  symbol,  V^,  according 
to  the  requirement  that  substitution  must  lead  to  an  identity, 
6  also  will  satisfy.     For  substituting  6, 
±5=:3-(±2) 

From  the  possibilities  as  to  sign  here,  and  one  sign  has  as 
good  a  right  as  another  to  be  chosen,  can  we  make  an  identity 
out  of  this?  By  taking  -y/  to  indicate  +  on  the  left  side, 
and  —  on  the  right,  we  have : 

5  =  3+2 

376.  The  point  to  be  noted  is  that  without  verifying  we 
should  not  have  found  precisely  which  root  would  satisfy  under 
the  assumption  of  §  375  as  to  the  sign  of  ■\/~. 

In  solving  radical  equations  verify  all  results  and  reject 
those  which  do  not  satisfy  the  original  equation,  on  the 
assumption  that  \/~always  denotes  the  positive  value  of  the 
root. 

Under  this  agreement,  which  is  convenient,  but  arbitrary, 
some   radical   equations   have   no   solution.     For  example. 

Solve  Vx^-m  =  7  +  Vx'  -  9  (1) 

Squaring  and  simplifying, 

a;2  -  37  =  V(a:2-lGHx2-9)  (2) 

Squaring  again  and  simplifying 

a:^  =  25,  or  x  =  =*=  5 
Substituting  in  (1),  vnder  the  arbitrary  agreement,  we  have 

3=7-|-4,  which  is  absurd. 
No  solution  is  possible  under  Ike  agreement. 
Setting  aside  the  agreement  and  substituting, 

±3=7±4_ 
Choosing  the  positive  value  ol  the  V   on  the  left,  and  the  negative 
value  on  the  right,  we  have 

3  =  7  —  4,  an  identity. 
Hence,    without  the   agreement,   a  solution   is  possible, 
though  the  radicals  must  be  given  whatever  algebraic  sign 
will  lead  to  an  identity. 


CHAPTER  XXIII 
QUADRATIC  EQUATIONS 

377.  A  quadratic  equation  is  an  equation  of  the  second 
degree  in  the  unknown  number.     For  example, 

x^+5x  =  20,  4:r2  =  36,  and  2x'--4x  =  5a, 
are  all  quadratic  equations. 

In  determining  the  degree  of  an  equation  it  is  assumed  that 
the  equation  is  first  reduced  to  its  simplest  form. 

378.  The  constant  term  in  a  quadratic  equation  is  the  term 
that  does  not  contain  the  unknown  number. 

Some  quadratic  equations  contain  only  the  square  of  the  unknown 
number;  others  contain  both  the  square  and  the  first  power  of  it.  Hence 
there  are  two  kinds  of  quadratic  equations. 

379.  A  pure  quadratic  equation  is  an  equation  that  does 
not  contain  the  first  power  of  the  unknown  number.     Thus, 

3a;2=108,  x^-\-2x  =  2x''+2x-m,  4x2  =  36a. 

380.  An  affected  quadratic  equation  is  an  equation  that 
contains  both  the  first  and  second  powers  of  the  unknown 
number.     Thus, 

3x2-f5a;  =  15,  x^-ix  =  S,  x^-ax=b. 

Pure  quadratics  are  also  called  incomplete  quadratics,  and  affected 
quadratics  are  called  complete  quadratics. 

THE   GRAPfflCAL   METHOD   OF   SOLUTION 

381.  The  Graphical  Solution.  The  normal  form  of  the 
pure,  or  incomplete  quadratic,  is  a;^  — a  =  0. 

Exercise  164  —  Graphing 
We  shall  now  graph  x^  —  a,  for  a  =  9,  a  =  4,  a  =  0,  and  a  =  —  4. 

282 


QUADRATIC   EQUATIONS 

1.  Graphing  x'  — a  for  a  =  9,  or  graphing  x' 
calculate  and  locate  the  points : 

x=     0,       1,       2,  3,  4,     5,    -1,    -2,    - 
a;2_9=_9^    _8^    _5^  0,  7,   16,    -8,    -5, 


283 

9,  we 

first 

3, 

-4, 

-   5 

0, 

+7, 

+  16 

Draw  a  smooth  curve  (1) 
through  these  points.  Recall  that 
^2_9_Q  asks:  ''What  is  x  where 
.T2-9is0?"  or  ''What  is  x  where 
the  curve  crosses  the  horizontal  V 
The  answer  is  readily  seen  from 
the  figure  to  be  +3  and  —3. 

Hence,  the   roots    of    x^  — 9  =  0 
are    +3  and   —3.     These  substi- 
tuted   in    x'^  —  9  =  0    are    seen   to     ^" 
satisfy  it. 

2.  Graphing  x-  — a  for  a  =  4,  or 

graphing  .r'-  — 4,  we  calculate  and 
plot  the  points : 

2=1  vertical   space 

x=  0,  1,2,  3,  4,  5,-1,-2,-3,  -4,  -5 
x2-4=  -4,  -3,  0,  +5,  +12,  +21,  -3,  0,  +5,  +12,  +21 
and  draw  a  smooth  curve,  like  curve  (2),  through  the  points. 
This  curve  is  of  the  same  form  as  curve  (1),  but  is  simply 
raised  upward  5  units.  The  a;-values  of  the  crossing  points 
are  here  +2  and  —2,  which  are  the  roots  of  x^  — 4  =  0. 

3.  Similarly,  graphing  curve  (3)  for  x-  — a,  for  a  =  0,  or 
graphing  the  curve  for  x^,  the  required  curve  is  drawn  through 
the  following  calculated  and  plotted  points : 

x  =  0,     1,    2,    3,      4,       5,     -1,     -2,     -3,       -4,       -5 


_j 

_j 

\ 

J 

h 

viAt 

t7$'7 

^^|i\i«. 

Hz'} 

fi 

f%  o> 

^'m 

J 

jlrj 

IJUi 

\i\\ 

n 

t' 

)             ®/ 

n 

\            i^i^ 

^Y        Ywl 

f 

\l\       ^Ja  A/i 

V 

i  \  g5  Jf  J ''' 

\ 

xfvt^i  Ahl 

>       \ 

\%^  jjf'i//" 

^ 

.-^-'i  q^  +  /2  -y^ 

^ 

\      |<4  rX  1/ 

~\  -■^      ~o~ 

\rf   J/_ 

~TSp^ 

1    '" 

Scale 
horizontal  epact 


25 


x2  =  0,     1,    4,    9,     16,    25,        1,        4,        9,        16, 

Here  there  is  but  one  x-value  of  the  crossing-,  or  rather 
touching-point  with  the  horizontal,  viz. :   0. 


284  ELEMENTARY  ALGEBRA 

Because  there  were  two  crossing-points  as  the  curve  moved 
upward  so  long  as  it  crossed  the  horizontal,  we  say  there  are 
two  equal  O's  here.  In  reahty  there  is  only  the  root  0,  because 
-fO  and  —0  are  the  same  point. 

4.  Graphing  x^  — a  for  a=  —  4,  or  graphing  x^+A,  we  cal- 
culate and  plot  the  points: 

x=     0,      1,      2,       3,       4,       0,-1,-2,-3,-4,-5 

0:2-1-4=  +4,  +5,  +8,  +13,  +20,  +29,  +  5,  +  8,  +  13,  +  2(),  +  29, 

and  draw  the  smooth  curve  (4)  through  them.  The  curve 
being  4  units  higher  than  curve  (3)  does  not  touch  the 
horizontal  at  all.  There  are  no  crossing-points  and  the 
algebraic  way  of  saying  this  is  to  say  the  roots  are  imaginary. 
We  shall  see  later  that  the  roots  are  +2\/—  1  and  — 2\/  — 1. 

382.  We  see  then  that  a  pure  quadratic  in  general  has  two 
roots  that  are  numerically  equal  but  of  opposite  signs,  but 
that  if  the  graph  of  the  first  member  just  touches  the  horizon- 
tal there  is  but  one  root,  viz.,  0.  If  the  graph  does  not  cut 
the  horizontal,  there  are  no  real  roots. 

But  since  two  results  are  found  by  solving 
x'^=  —a 
i.e.,  x=  \/  —  a,  and  x=  —  ^—a, 

we  say  that  if  the  graph  lies  entirely  above  the  horizontal,  there 
are  two  roots,  one  positive  and  the  other  negative,  and  both 
imaginary. 

SOLVING    QUADRATICS   BY   FACTORING 

383.  The  solution  of  quadratic  equations  by  factoring, 
given  in  §  215  and  on  page  164,  should  be  reviewed  here. 

This  is  not  a  general  method,  for  it  is  limited  to  those 
equations  the  first  members  of  which  are  readily  factored. 

A  pure  quadratic  equation  which  is  reducible  to  the  form 
a:^  —  a  =  0  is  readily  solved  by  factoring. 

When  reduced  to  this  form  it  is  evident  that  the  first 


QUADRATIC   EQUATIONS  285 

member  is  the  difference  of  two  squares.     For  example, 

{x-2a)  {x-\-2a)=Q  (.c- Vs)  (.c+ V5)=0 

x  =  2a,  and  -2a  x=V^,  and   -  V5 

Verify  that  the  values  found  for  x  are  solutions  by  sub- 
stituting them  in  the  original  equations. 

Exercise  166 
Solve  the  following  by  factoring  and  verify: 

1.  {x-iy  =  5-2x  2.  (.'r+3)--6(x+3)=9 

3.  4x'-\-9  =  x^-\-m  4.  x-\-Vx^+2'\/T^=l 


6.  2a;2-«  =  x2+3a  6.  Vx-{- y/¥^^^=  Va+x 

^^±n     x-n^  \/2x'~-\-l-\/2x^-l 

'  ^^     x-\-n~  '  ^2x^-\-l-\-\/2x''-l^^ 

^    x+2  .  x-2     „i  ^^      / 6 


a:-2  '  a:+2       -*  v  x  -r-ii-.t.- y^2_ii 


^^    x-t-4  ,  a:-4     ,7  ^„  /»  ,      /^ — r-^ 

11.  :ri-5+z — o  =  l8  12.     ^  ^^     _  =x4- v^  — 13 


5  5  1  a-H-y^''^x4-cr,  .g-ft 

3  — X     3+x       ^  '  x^— n^     x-\-n     x—n 


384.  Some  affected  quadratics  may  be  solved  in  a  similar 
manner  by  factoring.     For  example, 

x2-4a:-12  =  0  10x2-llx+3=0 

-      (a:-(5)(x+2)=0  (5a:-3)(2a;-l)  =0 

a:  =  6,  and  —2  ^=f,  and  ^ 

Substitute  these  values  of  x  in  the  given  equations  and 
verify  that  they  are  the  correct  roots. 


286  ELEMENTARY  ALGEBRA 

Exercise  166 
Solve  the  following  by  factoring  and  verify : 
1.  a;2+llx-26  =  0  2.  4x2-12a;=-9 

3.  2a^2_5^_i2  =  0  4.  6x2+lla;=-4 

.  6.  3a;2-7a;-20  =  0  6.  x2-20a:=-51 

385.  Some  equations  of  a  higher  degree  than  the  second 
may  be  solved  by  factoring.     Observe  the  following: 

x^—x-  =  l2x  x^— X-  — 4x-f  4  =  0 

x(x-4)  (x+3)=0  (x-2)  (x+2)  (a;-l)=0 

x  =  0,  4,  and  -3  x  =  2,  -2,  and  1. 

Substitute  these  values  of  the  unknown  in  the  given 
equations  from  which  they  were  found,  and  verify  that  they 
are  the  correct  roots. 

Exercise  167 
Solve  the  following  by  factoring  and  verify: 
1.  a;3+8x2-9x  =  0  2.  x^-{-dx^-x  =  5 

3.  x'^+ab-ax-bx  =  0  4.  x^-5x^-\-4:  =  0 

6.  x^-{-x^-4.2x  =  0  6.  x^+ax+bx-\-ab  =  0 

7.  x^-\-5x-Qx^  =  0  8.  6.T--49x=-8 
9.  x{x''-l)-2{x-\-l)=0         10.  x^-\-x'-'SOx  =  0 

11.  x^+7x'-7  =  x  12.  x{x''-4)-S(x-2)=0 

13.  6x2-f3x-18  =  0  14.  x^-x^-{-9  =  9x 

16.  (x-2)2-4(x-2)4-3  =  0  16.  x^-{-5x^-Qx  =  0 

17.  6x2+17x=-5  18.  {x^-x-2){Sx~--x-2)=0 
19.  6x2-5^-21=0  20:  .T^-17.'c2+16  =  0 


QUADRATIC   EQUATIONS  287 

SQUARE   ROOT   METHOD    OF   SOLUTION 

386.  A  pure  quadratic  is  solved  by  i^educing  it  to  tlie 
normal  form,  x-  =  a,  and  taking  the  square  root  of  both 
members. 

Root  Axiom.  Equal  principal  7'oots  of  equal  number's  are 
equal. 

Extracting  the  square  root  of  both  members, "we  have: 
x=  =t:  -y/a 

The  double  sign  belongs  to  the  unknown  number  as  well  as 
to  the  second  member,  but  x  =  =*=  \^a  is  the  same  as  —  a;  = 
=1=  \/a.  For  this  reason  the  double  sign  is  used  before  the 
second  member  only. 

A  pure  quadratic  equation  has  two  roots  numerically  equal, 
one  positive  and  the  other  negative. 

For  example, 

X-  =  2o,  x2  =  8,  x^=  —  5,  have  the  roots : 

Since  the  square  root  of  a  negative  number  is  imaginary, 
we  observe  that  when  a  is  negative,  both  roots  are  imaginary. 

All  this  was  shown  more  clearly  in  §  381  by  the  aid  of 
the  graphs. 

Exercise  168 

Solve  by  the  square  root  method : 

1    ^_  .  _i_=5  2    _1 3_^i 

5  .         a 


3.  \/^+5='    4.     ^.^=\/x-a 

■\/x  —  o  \/x-\-a 

387.  Any  complete  quadratic  equation  may  be  reduced  to 
the  normal  form, 

ax2+bx+c  =  0, 

a,  b,  and  c  denoting  any  real  numbers,  positive  or  negative, 
integral  or  fractional,  though  a  may  not  be  0. 


288  ELEMENTARY   ALGEBRA 

Since  any  complete  quadratic  may  be  reduced  to  this  form, 
it  is  called  the  general  quadratic. 

To  apply  the  square  root  method  of  solution,  the  first  member 
must  be  made  a  square.  For  this  purpose  the  form  of  the 
equation  is  changed  to : 

ax~-\-hx=  —c 

388.  The  process  of  making  the  first  member  of  a  quadratic 
equation  a  square  is  called  completing  the  square. 

The  value  of  a  in  the  general  quadratic,  ax--{-hx-\-c,  may 
be  1 ,  or  it  may  be  any  number  greater  than  1 . 

TO    COMPLETE   THE    SQUARE    WHEN    a    IS    1 

389.  Consider  the  arranged  trinomial  -square, 

Two  of  the  terms  are  squares  and  the  other  term  is  the 
product  of  three  factors,  viz.:  The  factor  2,  the  square  root 
of  the^rs^  term,  and  the  square  root  of  the  last  term. 

The  binomial  x^+2cx  represents  the  sum  of  the  first  and 
second  terms  of  any  arranged  trinomial  square.  Dividing  the 
second  term,  2cx,  by  twice  the  square  root  of  the  first  term, 
i.e.,  by  2x,  the  quotient  is  c,  which  is  the  square  root  of  the 
missing  term.  Adding  c^  to  x^-\-2cx  will  therefore  complete 
the  square. 

390.  Rule. —  Reduce  the  equation  to  the  general  form  and  add 
to  both  members  the  square  of  half  the  coefficient  of  x. 

To  make  the  first  member  of  x^  — 6x  =  7  a  square,  we  must  add  9  to 
both  members,  thus  obtaining: 

a-2-6x+9  =  16 
By  the  root  axiom,  .r  —  3  =  =*=  4 

Whence,  x  —  T,  and  —1 

Substitute  these  in  the  given  equation  and  verify. 
Carefully  observe  the  following  important  truth : 


QUADRATIC   EQUATIONS  289 

391.  The  sum  of  the  two  roots  is  the  coefficient  of  x  with 
reversed  sign.  The  product  of  the  two  roots  is  the  constant 
term  of  the  equation  in  the  general  form. 

To  the  teacher:  Require  pupils  to  test  or  verify  the  roots  of  all 
quadratic  equations  by  reference  to  the  foregoing  principle. 


For  example,  solve 

a:2-3x-18=0. 

x'--3x  =  18, 

x2-3x+f  =  8i 

.        ^-f=-f 

x  =  6,  and 

3 

The  sum  of  the  roots  is  3,  the  coefficient  of  x  with  reversed 
sign;  and  their  product  is  —18,  which  is  the  constant  term. 

Again,  solve  x-  —  6a:  + 1 2  =  0 

a:2_6x=-t2 
x2-6x+9=-3 

x-3=±V^ 
x  =  3±V^ 
The  sum  of  the  roots  is  6,  the  coefficient  of  x  with  reversed 
sign;  the  product  is  12,  which  is  the  constant  term. 

Exercise  169 

Solve  by  completing  the  square  and  verify : 

1.  x2-f-10x=-21  2.  ?/-4?/-117  =  0 

3.  n2-14n=-24  4.  .?/-6?/-160  =  0 

6.  x2-12a:=-32       .  6.  7/-2?/-143  =  0 

7.  /i2+lln=-24  8.  x2-3a:-180  =  0 

TO    COMPLETE   THE    SQUARE   WHEN   a   IS   NOT   1 

392.  Observe  the  following  solution  of  2x2  — 6x  — 5  =  0,  jj^ 
which  the  coefficient  of  x^  is  greater  than  1. 

Dividing  through  by  2,  x^  -  3x  -  f  =  0  ( 1 ) 

Transposing  the  |^,  x^  —  3a:  =  f 

Completing  the  square,  x-  — 3a:-}-f  =f -f  f 
By  the  root  axiom,  x  — f  =  ±  i\/l9 

Hence,  j;=f=*=i\/lO 


290  ELEMENTARY   ALGEBRA 

The  sum  of  the  roots  must  be  the  negative  coefficient  of  x 
in  the  equation  in  which  the  coefficient  of  x-  is  1  [i.e.,  in  (1)], 
and  the  product  of  the  roots  must  be  the  constant  term  in  the 
same  equation. 

The  sum  of  the  roots  is  3,  the  coefficient  of  x  with  reversed 
sign;  the  product  is  — -|,  which  is  the  constant  term. 

Exercise  170 

Solve  the  following  and  verify : 

1.  x2-168=-2x  2.  2x^+Sx-U  =  0 

3.  Zx^-10x=-S  4.  3a;2-h4x-39  =  0 

5.  2/2-120= -2?/  6.  2x2+7a:-39  =  0 

7.  8a:-a;2=-180  8.  n2-lln-60  =  0 

9.  a:2-16x=-60  10.  if -\-loy-d^  =  Q 

11.  3x2-33= -2a;  12.  x2-13a;-30  =  0 

13.  w2-lln=-30  14.  3a;2-f-x- 200  =  0 

16.  3x2-95= -Ix  16.  ?/2-ll?/-|-28  =  0 

393.  To  avoid  fractions,  first  multiply  both  members  of  the 
equation  by  four  times  the  coefficient  of  x^. 

For  example,  to  solve :  2x-  —  7x  —  1 5  =  0 

Multiply  by  8,  16x2-56x  =  120, 

Dividing  b^x  by  twice  the  square  root  of  \^x-,  the  quotient  is  7. 
Squaring  7  and  adding, 

16x2-56rc+49  =  169 

By  the  root  axiom,  4a: —7  =  =*=  13, 

Whence,  x  =  b  and   — f. 

If  the  coefficient  of  x-  in  the  given  equation  is  made  1,  the  coefficient  of 
X  is  —  1^  and  the  constant  term  is  — V^. 

The  sum  of  the  roots  is  -g-,  the  coefficient  of  x  with  reversed  sign;  the 
product  is  —  V",  which  is  the  constant  term.     This  checks  the  work. 

Observe  that  the  number  added  to  complete  the  square  is 
the  square  of  the  coefficient  of  x  in  the  given  equation. 


QUADRATIC   EQUATIONS  291 

Exercise  171 
Complete  the  square,  solve  and  verify : 
1.  3a:2-7x=-2  2.  2x2~5a;-42  =  0 

3.  x'^-12=-4x  4.  3a;2-2a;-40  =  0 

5.  4x'--7x=-3  6.  5r2-14r+8  =  0 

7.  7w'+Qm=-n  8.  31*2 -f-9ii- 30  =  0 

9.  2n2-5=-3n  .  10.  3?/2- 101/4-3  =  0 

11.  x2+6a;=-25  12.  ?/-10i/+21=0 

13.  3<2-2=-5^  14.  2s2+7s-22  =  0 

15.  rt2-j_8a=-21  16.  52-125-45  =  0 

SOLUTION   BY   FORMULA 

394.  The  equation  ax2H-bx+c  =  0  may  be  taken  to  repre- 
sent, or  typify,  any  quadratic  equation,  in  which  all  terms 
have  been  transposed  to  the  first  member,  the  a:2-terms  being 
combined  into  a  single  term,  as  also  the  x-terms,  and  the 
constant  terms. 

The  solution  of  ax^-]-hx-\-c  —  0  gives  a  formula,  or  short- 
hand law  for  writing  the  roots  of  any  equation  of  that  form. 

Completing  the  square  and  solving, 

-b=bVb2-4ac 
x  = 2^ 

2a 

This  is  the  formula  for  writing  the  roots  directly  without 
completing  the  square.  It  is  the  final  result  that  is  always 
arrived  at  by  completing  the  square,  and  it  may  always 
be  written  down  at  once. 


202  ELEMENTARY  ALGEBRA 

Notice  there  are  tvw  roots,  viz.: 

-b  +  \/b'-4ac         b  ,  \/b2-4ac 


Xi  = 


2a  2a  2a 


_  -b-\/b^-4ac_      b      \/b^-4ac 
^^  2a  2a  2a 

Solve  the  following  by  the  formula: 

1.  a:2-10a:-24  =  0        

a;  =  5±\/25+24 
a:  =  12  and   -2 

2.  2x2 -*13a;+ 15  =  0 

X 

X 


—  13.7 


a:  =  o  and  1^ 

By  the  use  of  this  formula  write  by  inspection  the  roots  of 
the  equations  at  the  end  of  Exercise  171. 

TO  FIND  APPROXIMATE  VALUES  OF  ROOTS  OF  QUADRATIC  EQUATIONS 

395.  Observe  the  following  process  for  calculating  approx- 
imate roots: 


(1) 

(2) 

:2-9x+16  =  0 

x2- 

-12a:+25  =  0 

-9a:4-(|)2=-V— 16=\'- 

0:2- 

-12a:+62  =  62-25  =  ll 

x-^  =  ^\y/\7 

•   a:-6=+JVTi 

x-  =  4.5±2.062- 

a:  =  6=±=l.658+. 

a:  =  6.562 - 

a;  =  7.658 + 

and  2.438  4- 

and  4.342- 

Observe  in  each  case  whether  the  sum  of  the  roots  equals 
the  coefficient  of  x  with  reversed  sign. 

Exercise  172 

Find  the  approximate  roots  to  two  places  of  decimals,  of 
the  following: 

1.  a:2-3:r-8  =  0  2.  a:2-5x+3  =  0 


QUADRATIC   EQUATIONS  293 

3.  x2+7x-22  =  0  4.  a;2-8x-38  =  0 

5.  a:2-f  lla:+27  =  0  6.  a;2-6a:-35  =  0 

7.  a;2-10.T+23  =  0  8.  x^-{-4:X-U,92==0 

9.  x2_2x-5.76  =  0  10.  a:2+2x- 20.78  =  0 

11.  .T2-5.2a;  +  5.76  =  0  12.  .t^- 9.65a; +10.5  =  0 

13.  j2- 11. 05a; -96.6  =  0  14.  a:2-22.55x+96.6  =  0 

EQUATIONS   IN    QUADRATIC   FORM 

396.  An  equation  is  in  the  quadratic  form  when  it  contains 
but  two  powers  of  the  unknown  number,  the  exponent  of  one 
power  being  twice  that  of  the  other.  Show  that  the  following 
are  in  the  quadratic  form: 

2x'-3x''  =  S,  2a;+4a--  =  13,  and  3a;' -2a:' =4 

These  equations  which  are  said  to  be  m  quadratic  form  may 
be  reduced  to  the  form 

ax2"+bx"+c  =  0, 
and  they  may  be  solved  by  any  of  the  methods  for  solving 
complete  quadratics. 

The  first  solutions,  however,  are  the  values  of  x",  that  is, 
the  values  of  x  with  half  the  larger  exponent. 

Evolution  and  involution  must  then  be  applied  to  both 
members  of  the  equation  to  find  the  values  of  x. 

Solve  the  following  equations  that  are  in  quadratic  form: 

(1)      .T^- 13x2+36  =  0  (2)       2a:+3V7=27 

(.r^-4)(.r2-9)=0  16j;+24x'  =216 

.t2  =  4  and  9  1 6.r + 24x'  +9  =  225 

.r==t2and   ±3  4x-i+3=±15 

a;' =3  and   -f 
.T  =  9  and  -V" 

In  verifying  these  values,  remember  that  in  this  particular 
example  the  square  root  of  V"  i^  ~f >  because  —  f,  not  +f , 
was  squared  to  give  -\^ . 


294                           ELEMENTARY  ALGEBRA 

Exercise  173 

Solve  the  following  equations: 

1.  x4+4a;2-45  =  0  2.  a;^+x*-30  =  0 

3.  x'-5x'-2A  =  0  4.  2v^4-3\/x  =  6 

6.  rc+6V^-20  =  0  6.  a;^+4a:^-5  =  0 

7.  x4-5x2-36  =  0  8.  \/x-3v^  =  28 
9.  a:'+2x^-8  =  0  10.  2x3H-5-v/x3  =  7 

11.  x^+4x2-32  =  0  12.  0^6+2x3-80  =  0 

13.  x-5V^-14  =  0  14.  4x^+x«-39  =  0 

397.  Some  expressions  are  in  quadratic  form  with  reference 
to  a  compound  expression,  such  for  example  as, 

(x-\-2y-{x-\-2)  =  12  and  a:+3+2\/^+3-3  =  0 
These  equations  may  be  solved  by  factoring,  the  first  one 
for  {x-{-2)  and  the  second  one  for  \/^H~3. 

Exercise  174 

Solve  the  following  by  factoring: 

1.  x-8-V^^  =  20  2.  (x-2)2-3(a:-2)  =  10 

3.  a;+6-2\/i+6  =  8  4.  (0:2-5)2-4x2+20  =  77 

6.  x+4+(x+4)'  =  20  6.  (x2+8)2-5x2-40  =  84 

398.  Some  equations  may  be  put  in  the  quadratic  form 
by  adding  a  number  to  both  members.     For  example, 

x2-4x+\/^'-4x+12  =  8 

may  be  put  in  quadratic  form  by  adding  12,  thus: 

x2-4x+12+\/^'-4x+12  =  20 

This  is  in  the  quadratic  form  with  reference  to  .r^— 4^+1 2. 

By  factoring,         ^     vx^  — 4a; +  12= —5  and  4 
Squaring,  .t-  —  4a;  + 1 2  =  25  and  1 6. 

andx2-4a:-4  =  0 
The  last  two  equations  are  ordinary  quadratic  equations. 


QUADRATIC   EQUATIONS 
Exercise  175 
Solve  the  following  equations  in  quadratic  form : 
1.  3x^+50:2-8  =  0 


295 


3.  x^+5x^+6  =  0 

6.  V^-3\/^=21 

7.  3^-5x--2^  =  0 
9.  a:-^-5a:-^+4  =  0 

11.  2x-^-a;-^-45  =  0 
13.  (x-l)^  +  (x-l)^  =  2 
15.  (x-5)2-x+5  =  110 


2.  x2-7a;-V^2_ 
4.  x^  —  Qx—  \/4x2 


7x+l  =  5 


24x  =  8 


6.  x2+V^2_5^_|_3^5^^3 
8.  x''-2x-\/9x'--lSx  =  4: 
10.  (x2+3)2-5(a:2+3)  =  14 
12.  r'-6x-3  =  2\/^^-6x 
14.  x^-5x+  \/4x^ -  20x  =  48 
16.-\/2x2+14x+2  =  x2+7x-3 


GRAPHICAL   SOLUTION   OF   QUADRATICS 

399.  The  graphical  solution  of  quadratic  equations  makes 
the  meaning  of  the  roots,  and  the  possibility  of  solutions, 
somewhat  clearer. 

To  solve  graphically  the  equation 

x2-6x+8  =  0. 
First  graph  the  function  x^  — 6x+8  for  the  values: 

x=-l,      0,       1,  2,      3,  4,      5,      6,        7,  etc., 
x2-6x+8=+15,  +8,  +3,  0,  -1,  0,  +3,  +8,  +15,  etc.. 

Plotting  these  points  and 
connecting  them  as  in  the  figure 
we  have  the  graph  of  .t^  — 6x+8. 
To  ask  for  the  values  of  x  that 
give  x^  — 6a:+8  =  0,  is  to  ask 
what  are  the  x-values  of  the 
crossing -points  of  the  graph 
over  the  horizontal.  Clearly 
these  values  are  x=+2  and 
x=+4. 

The  curve  of  the  figure  is 
called  a  parabola  and  any  quadratic  like  x^-{-px-\-q  always 
gives  a  'parabola  for  its  graph. 


— i: V — 

:::^:::i:;^i:: 

:i:i^:ii:^Ei:: 

:i::Vi|iii: 

±::iffi:iii:'' 

Y' 

Scale 

1  •■  1  horizontal  space 

2  "1  vertical   space 

Graph  of  rc2-6a;+8 


296 


ELEMENTARY  ALGEBRA 


Graph   of   x'+ax    +  12 

for    a  =+8     (I) 

a  =  -  S     (2) 

a  =  -i-7     (3) 

a^-7     (4) 

Scale 

2—1  vertical  space 


400.  This  figure  gives  the  graphs  of  four  quadratics  ob- 
tained by  keeping  the  constant  term  equal  to  + 12  and  chang- 
ing the  coefficient  of  the 
x-term  only. 

The  quadratics  x^ + 7.r + 1 2 
and  ^2  —  7x+ 12  give  the  same 
shape  of  curve;  either  being 
turned  over  the  vertical  axis 
gives  the  other.  The  same  is 
true  of  the  graphs  of  x"-\-Sx 
-\-l2'dndx--Sx+l2. 

This  may  be  expressed  by 
saying  that  reversing  the  sign 
of  the  coefficient  of  x  in  the 
quadratic,  turns  the  graph 
over  around  the  vertical  axis. 

The  roots  of  such  pairs  of  quadratics  are  numerically  equal, 
but  of  opposite  signs. 

Give  the  roots  from  the  figure  for  quadratic  equations 
made    by    putting   each 
of    the    four    quadratic 
trinomials  equal  to  0. 

All  four  of  the  graphs 
go  through  the  point 
+  12  on  the  vertical. 

401.  This  figure  shows  ^' 
the  graphs  of  quadratics 
all  of  which  have  the 
constant  term  —12. 
Compare  the  graphs  of 
the  pairs: 


Y 

\  i  R~r             N  /  J  t    i\ 

I  rrV"         JLJiLA- 

1  lit         -T/fr  p- 

"^trW""  i"  "l^r  "fi~ 

A    jA  V-ju       _j y  I'^Ih    /y 

s  \     <^^\    \w\^                       /    /    /         /-v |_ 

"^  1    1  \  \1  \            1/  Mf     7^ 

"$Il     \  \     f  !  //'^^  ^ /  "^  ^ 

Y    \  Y  A        /[/  /     / 

3             ?  A          Lri-''          t- 

\    W\    t/\ /    >i' 

V  \ \i\  y/iZ^  J. 

^v-^^^^j: 

\    42a_    2   it 

"    V.^f  Vvf  1 

x2+a:-12 

Graph  of  x'-¥  aa>  -  12 

a;2-x-12, 

for  o  =  +^ 
a  =  -4 

a=  +i 

x2+4a:-12 

a=  -; 

x2-4x-12. 

Scale 
2  -=  1  vertical  space 

QUADRATIC   EQUATIONS 


297 


Reversing  the  sign  of  the  x-term  again  is  seen  to  turn  the 
graphs  over.     Through  what  point  do  all  these  graphs  go? 
Read  from  its  graph  the  roots  of  — 

^  a;2-4x-12  =  0  x2-a;-12  =  0 

402.  The  figure  shows  tliat  the 
effect  of  changing  only  the  constant 
term  is  to  hold  the  curve  of  the  same 
shape  and  to  raise  it  by  just  as 
much  as  the  constant  term  is 
increased. 

The  curve  crosses  the  horizontal 
in  two  points,  giving  two  real  roots, 
^  until  it  just  touches  the  horizontal. 
Then  the  two  roots  coalesce  into  one. 
As  the  curve  rises,  it  ceases  to  touch 
the  horizontal  and  the  roots 
become  imaginary,  as  the  following 
algebraic  solution  of  x^  — 2x4-5  =  0 
will  show. 

Solving  x^  — 2x+5  =  0,  which  is 
the  same  as — 


X' 


r          n 

_i 

(^' 

A 

T 

% 

//! 

^v\v 

)l  jv^ 

\\iv 

yj// 

n]\V\ 

JjIL'^ 

'i\i^\ 

tiff 

\''W 

Sj'j 

\\  \l\ 

1  ih  1 

~^Wv~ 

-4ffi-r 

—  WM 

W" 

""■rVr 

^Xxj 

\-k 

^^  1 

4^ 

y 

^ 

J 

r^ 

.■^ 

Graph  of   x- 
for  6=  -  8, 
6=  -5. 

6=  +  5. 


-  2x  4-  6 
curve  (I) 
curve  {2) 
curve  (S) 
curve  d) 


Scale 
2=1  vertical  space 


x2-2a:+l=-4 
(a:-l)2=-4 

x—l=  =*=\/^,  or  =t2-\/^, 
we  obtain,  x=l=i=2^/— 1 

These  roots  are  imaginary,  since  they  contain  \/  — 1. 

Thus,  failure  of  the  graph  either  to  cut  or  to  touch  the 
horizontal  indicates  that  imaginary  roots  are  present. 

When  the  graph  touches,  or  cuts,  the  horizontal  how  could 
the  factors  of  the  first  number  of  its  equation  be  read  from 
the  graph? 


298  ELEMENTARY  ALGEBRA 

Exercise  176 

Solve  the  following  quadratic  equations  graphically : 
1.  a:2-3x~10  =  0  2,  a;2+3a;-10  =  0 

3.  x2-5x-6  =  0  4.  x2+a:-20  =  0 

6.  x2-x-20  =  0  6.  x2+5x  =  0 

CHARACTER  OF  THE  ROOTS  OF  QUADRATIC  EQUATIONS 

403.  The  character  of  the  roots  of  any  complete  quadratic 
equation  is  determined  by  examining  the  solutions  of: 
ax2+bx+c  =  0 

In  this  discussion  it  is  assumed  that  a,  h,  and  c  are  real 
numbers,  a  is  greater  than  zero,  and  b  and  c  are  either  posi- 
tive or  negative. 

Denoting  the  roots  by  n  and  ro,  we  have  the  values: 

— b  +  \/b2— 4ac  —  b  — \/b2  — 4ac 


2a  2a 

The  nature  of  the  two  roots,  as  real  or  imaginary,  rational 
or  irrational,  depends  on  the  value  of  6^  — 4ac. 

The  expression  h^—Aac  is  called  the  discriminant  of  the 
roots. 

404.  Observing  the  formulas  for  n  and  r2,  it  is  evident 
that: 

1.  When  the  discriminant  is  a  square  the  roots  are  real, 
rational,  and  unequal. 

2.  When  the  discriminant  is  equal  to  zero  the  roots  are  real, 
rational,  and  equal. 

3.  When  the  discriminant  is  a  positive  number  not  a 
square  the  roots  are  real  and  conjugate  surds, 

4.  When  the  discriminant  is  a  negative  number  the  roots 
are  conjugate  complex  numbers. 

A  complex  number  is  a  number  of  the  form  a-\-b\^  —  1,  a  and  b 
denoting  real  numbers. 

The  numbers  a+^v  —  1  and  a— 6V  —  1,  are  conjugate  complex 
numbers. 


QUADRATIC   EQUATIONS  299 

405.  It  follows  that  we  can  determine  the  nature  of  the 
roots  of  any  quadratic  equation  without  solving  it.  For 
example: 

3x2-7x-h2  =  0 

In  this  equation  6^— 4ac  =  25.  Since  25  is  a  square,  the 
roots  are  real,  rational,  and  unequal.     But  take  the  equation 

In  this  equation  b^—4ac=  —24.  Since  —24  is  a  negative 
number,  the  roots  are  conjugate  complex  numbers. 

Exercise  177 

By  the  use  of  the  discriminant  determine  the  nature  of 
the  roots  of  each  of  the  following  equations : 

1.  4a:2-7x+3  =  0  2.  a:2-7a;-8  =  0 

3.  2x2-4a^+2  =  0  4.  x^+6x-\-^  =  0 

6.  5x2-f-8x-2  =  0  6.  x2-3x-h5  =  0 

7.  7x2-5x+l=0  8.  x^-\-Zx+5  =  0 
9.  4x2-4x+l=0                                     10.  x^-dx-9  =  0 

11.  4a;2+6x-4  =  0  12.  a:2-5a:+8  =  0 

13.  For  what  values  of  n  will  2x^-\-nx-\-S  =  0  have  equal 
roots?     Irrational  roots? 

14.  For  what  value  of  a  will  ax'  —  l2x+Q  =  0  have  equal 
roots?     Imaginary  roots? 

15.  For  what  values  of  n  will  Sx^-{-2nx-\-S  =  0  have  equal 
roots?     Imaginary  roots? 

16.  For  what  values  of  c  will  Sa;^  — 10xH-c  =  0  have  equal 
roots?     Real  roots?     Imaginary  roots? 

17.  For  what  values  of  n  will  9x'^-{-nx-{-x-\-l=0  have 
equal  roots?     Find  the  corresponding  values  of  x. 


300  ELEMENTARY   ALGEBRA 

406.  By  dividing  both  members  of  the  general  quadratic 
equation,  ax^-\-bx-\-c  =  0  by  the  coefficient  of  x^,  the  equation 
becomes  of  the  form: 

x2-f2px+q  =  0 

in  which  p  and  q  are  positive  or  negative,"  integral  or  frac- 
tional, and  2p  is  any  coefficient  of  x. 

The  solutions  of  this  equation  are,  by  §  394  or  §  403. 

ri=  — p+Vp^  — q 


r2=-p-\/p'-q 

The  sum  of  the  two  roots  of  JC--|-2^x+^  =  0  is  —  2p,  the 
coefficient  of  x  with  reversed  sign. 

The  product  of  the  two  roots  of  X"+2px-\-q  =  0  is  q,  the 
constant  term  of  the  equxition. 

407.  The  two  foregoing  principles  enable  us  to  form 
quadratic  equations  with  given  roots. 

If  the  roots  of  a  quadratic  equation  are  —9  and  5,  the 
coefficient  of  x  is  4,  and  the  constant  term  is  —45.  The 
equation  then  is 

x2+4a;-45  =  0 

It  has  already  been  proved  §§  215,  384-5,  that  if  (xH-9) 
(x  — 5)=0,  the  roots  are  —9  and  5. 

Observe  that  the  known  numbers  in  (xH-9)(a;  — 5)  =0^,  are 
the  roots  of  the  equation  with  their  signs  reversed. 

TO   FORM   A    QUADRATIC   EQUATION    WITH    GIVEN    ROOTS 

408.  Rule. —  Subtract  each  of  the  roots  from  x  and  place  the 
product  of  the  two  remainders  equal  to  zero. 

The  equation  whose  roots  are  6  and  —7  is 

(a;-6)(x+7)=0,  or 
a;2-f-x-42  =  0. 


QUADRATIC   EQUATIONS  >  301 

Exercise  178 
Give  at  sight  the  equations  whose  roots  are : 


1. 

5  and  3 

2.  2  and  —5               3.   —5  and  —4 

4. 

7  and  2 

5.  Sand  -3               6.   -3  and  -8 

7. 

6  and  5 

8.  3  and  -7               9.   -7  and  -5 

10. 

8  and  9 

11.  9  and  -4             12.   -4  and  -7 

13. 

f  andf 

14.  I  and  -|            15.   -|  and  -| 

16. 

a-\-n  and  a—n 

17.   —a-\-\/7iand—a  —  \/n 

18. 

a  — 2  and  a+2 

19.   -2+\/3and -2-\/3. 

20. 

2a+l  and  2a-: 

L              21.   -3-\/5and -3-f-A/5 

22. 

3-2?:and3-h2i 

23.   -H-\/7and  -1-V7 

24. 

What  is  the  sum 

of  the  roots  of  x^^- a;  -  6  =  0?     What  is 

the  sum  of  the  roots  of  2a;^+12a:-f  1  =0? 

26.  If  a  in  the  general  quadratic  equation  is  5,  what  part 
of  b  is  the  sum  of  the  roots? 

26.  For  what  value  of  c  will  4^2— 16x4- c  =  0  have  equal 
roots?     Conjugate  surd  roots?       Imaginary  roots? 

27.  For  what  value  of  m  will  3x2  — mx— 48  =  0  h^ve  equal 
roots?     Conjugate  surd  roots?     Imaginary  roots? 

FACTORING   BY   PRINCIPLES    OF    QUADRATICS 

409.  The  method  of  factoring  quadratic  trinomials,  whose 
factors  are  rational,  has  already  been  explained.    (See  p.  163.) 

By  the  principles  of  quadratics,  quadratic  expressions 
whose  factors  are  irrational  may  be  factored.  For  example, 
To  factor  x2-8x+ 11. 

We  place:     x'-Sx  +  U^O 

By  §  390,  x  =  Aj-V5  and  A-Vl. 

Hence,  by  §  408  {x-4-V5)(x-i-\-V5)  =-x^-Sx-\-n._ 

The  factors  of  a:-  —  8x  + 1 1  are  a-  —  4  —  V  5  and  x  —  4  +  v  5. 


302 


ELEMENTARY   ALGEBRA 
Exercise  179 


Factor  the  following: 

1.  a2-4a+l 

4.  a2+6a-3 

7.  a2-2a+4 

10.  a-+5a— 1 


2.  x^-Hx-2 

5.  a;2+4a;-4 

8.  a:2+8a;-8 

11.  x'^-Sx+l 


3.  n2-6n+ll 
6.  n2-6n+13 

12.  /i2+9n+23 


PROBLEMS  IN  QUADRATIC  EQUATIONS 

410.  Since  quadratic  equations  have  two  roots,  a  problem 
whose  solution  involves  such  an  equation  apparently  has  two 
values  of  the  unknown  number,  or  two  roots. 

Both  roots  may  satisfy  the  equation,  but  only  one  of  them 
may  satisfy  the  conditions  of  the  problem.  Especially  is  this 
true  when  the  roots  are  surds  or  imaginary. 

In  solving  problems  that  involve  quadratics,  we  should 
examine  the  roots  of  the  equation  and  reject  any  root  that 
does  not  satisfy  the  requirements  of  the  problem. 


Exercise  180 —  Problems  in  Quadratics 

Solve  the  following  problems: 

1.  The  sum  of  two  numbers  is  42,  and  their  product  is 
416.     Find  the  two  numbers. 

2.  The  sum  of  the  squares  of  three  consecutive  numbers  is 
590.     Find  the  three  numbers. 

3.  A  rectangular  field  of  4  acres  is  12  rods  longer  than  it 
is  wide.     What  are  the  dimensions? 

4.  The  quotient  of  one  number  divided  by  another  is  7, 
and  their  product  is  2800.     Find  the  numbers. 

5.  If  the  sum  of  the  squares  of  three  consecutive  even 
numbers  is  980,  what  are  the  numbers? 


QUADRATIC   EQUATIONS  303 

6.  What  is  the  price  of  eggs  per  dozen  when  5  less  for  50^ 
increases  the  price  6^  a  dozen? 

7.  Find  two  consecutive  odd  numbers  the  sum  of  whose 
squares  exceeds  20  times  the  larger  number  by  94. 

8.  The  perimeter  of  a  rectangular  field  is  84  rods,  and  the 
area  is  432  square  rods.     Find  the  dimen'sions. 

9.  The  sum  of  two  numbers  is  24,  and  their  product  is 
139.     Find  the  numbers  and  prove  your  answer. 

10.  The  difference  between  two  numbers  is  16,  and  their 
product  is  1380.     Find  the  numbers. 

11.  The  perimeter  of  a  rectangular  field  is  114  rods,  and 
the  area  is  5  acres.     Find  the  dimensions. 

12.  Solve  the  formula  d  =  ^gf  for  t  and  g. 

13.  The  sum  of  two  even  numbers  is  48,  and  the  sum  of 
their  squares  is  1224.     Find  the  numbers. 

14.  The  sum  of  two  numbers  is  40,  and  their  product  is 
398 J.     Find  the  numbers.     Prove  your  answer. 

16.  The  sum  of  two  numbers  is  96,  and  their  product  is  18 
times  as  much.     Find  the  numbers. 

16.  Solve  the  formula  a^  =  ¥+c'^  for  b  and  c. 

17.  The  hypotenuse  of  a  right  triangle  is  9  feet  longer 
than  one  leg  and  2  feet  longer  than  the  other  leg.  Find  the 
three  sides  of  the  triangle. 

18.  At  15^  a  square  foot,  it  cost  $99  to  lay  a  parquet 
floor  in  a  room  whose  length  is  8  feet  more  than  its  width. 
Find  the  dimensions  of  the  floor. 

19.  The  dimensions  of  a  certain  rectangle  and  its  diagonal 
are  represented  by  three  consecutive  even  numbers.  What 
are  the  dimensions  of  the  rectangle? 


304  ELEMENTARY  ALGEBRA 

20.  A  carpenter  worked  30  days  more  than  lie  received 
dollars  per  day  for  his  labor  and  earned  $175.  How  many 
days  did  he  work  and  how  much  did  he  receive  per  day? 

21.  Two  numbers  differ  by  1.  The  square  of  their  sum 
exceeds  the  sum  of  their  squares  by  220.     Find  the  numbers. 

22.  There  are  32  sq.  yd.  in  a  rectangle  whose  length  is  18 
times  the  width.     Find  the  length  in  feet. 

23.  Find  two  numbers  whose  difference  is  6,  and  whose 
sum  multiplied  by  the  smaller  number  is  756. 

24.  Find  the  side  of  a  square  whose  area  is  doubled  by 
increasing  its  length  9  yd.  and  its  width  6  yd. 

26.  One  square  field  is  10  rd.  longer  than  another,  and  the 
area  of  both  is  1 108  sq.  rd.     Find  the  length  of  each. 

26.  Find  the  numbers  the  sum  of  whose  two  digits  is  13 
and  the  sum  of  the  squares  of  whose  digits  is  89. 

27.  The  number  of  square  inches  in  the  surface  of  a  cube 
exceeds  the  number  of  inches  in  the  sum  of  its  edges  by  1170. 
Find  the  volume  of  the  cube, 

28.  A  man  bought  a  piece  of  land  for  $4050.  He  sold  it  at 
$53  an  acre,  making  a  profit  equal  to  the  cost  of  16  acres. 
How  many  acres  did  he  buy? 

29.  A  merchant  sold  some  damaged  goods  for  $24  and  lost 
a  per  cent  equal  to  the  number  of  dollars  he  paid  for  the  goods. 
Find  the  cost  of  the  goods. 

30.  The  length  of  a  rectangle  exceeds  its  width  by  7  rd. 
If  the  dimensions  were  increased  5  rd.,  it  would  contain 
5  acres.     Find  the  dimensions  of  the  rectangle. 

31.  A  merchant  bought  lace  for  $100.  He  kept  30  yards 
and  sold  the  remainder  for  as  much  as  it  all  cost,  gaining  75<^ 
a  yard.     How  many  yards  did  he  buy? 


CHAPTER  XXIV 

SIMULTANEOUS  SYSTEMS  SOLVED  BY 
QUADRATICS 

411.  A  quadratic  equation  in  two  variables  (unknowns) 
is  an  equation  of  the  second  degree  in  the  variables  (un- 
knowns).    Thus,  X  and  y  denoting  variables, 

3a:'--2i/2+x-4?/+3  =  0,  x+Si/-y  =  9,  and  xy-lQ  =  x-y, 
arc  quadratic  equations  in  two  variables. 

412.  Two  or  more  such  equations  in  the  same  variables  are 
called  a  system  of  quadratic  equations.  If  all  equations  of 
the  system  can  be  satisfied  by  the  same  pair,  or  pairs,  of  val- 
ues of  the  variables,  it  is  called  a  simultaneous  system. 

Not  all  simultaneous  systems  of  quadratic  equations  can 
be  solved  by  elementary  algebra.  In  fact  the  solution 
generally  leads  to  biquadratic,  or  fourth  degree  equations. 

We  shall  consider  here  only  systems  containing  one  quad- 
I'atic  and  one  linear  equation. 

413.  Let  us  now  examine  the  meaning  of  the  solutions 
of  such  equations,  beginning  with  the  system, 

y  =  4:X  —  x~—l  (1) 

y  =  2x-l  (2) 

Equation  (1)  is  the  same  sls  x'^—4x-\-y-\-l=0. 
Calculating  from  equation  (1),  the   ^/-values   for   the   x- 
values  of  the  first  line, 

x=-l,-^,  0,  fi,+l,+2, +3, -|-3i+4, -h4i,+5,etc., 
y=  -6,  -3i,  -1,  +i  +2,  -f  3,  +2,  +f ,  -1,  -3^  -6,  etc., 
plotting  the  number-pairs  and  connecting  the  points,  gives 
the  curT^e  in  the  figure. 

305 


306 


ELEMENTARY    ALGEBRA 


~JI    ^ 

^ffyf\ 

r  yaw  \  q 

»5=l=^ 

fi---#i 

Y' 

Scale 
1     horizontal    t>pace 

1     vertical   e^pacu 


On  the  same  reference  lines,  graphing 
(2)  for 

'a:=+2,  -2 
i/=H-3,  -5 

x'\  I  rj^K/?."\|  I  '  \x  gives  the  straight  hne  marked  y  =  2x—l 
in  the  figure. 

The  solutions  sought  are  the  x-  and 
^/'distances  of  the  crossing-points  of 
the  graphs  of  (1)  and  (2). 

The  X-  and  y-values  must  he  so  paired 
that  both  numbers  of  each  pair  belong 
to  the  same  crossing-point. 
The  solutions  are :    x  =  0,  y=  —1,  and  x  =  +  2,  y=-\-S. 

414.  The  graph  of  (1)  is  a  parabola  and  any  two-letter 
equation  of  the  second  degree  with  only  one  variable  raised  to 
the  second  power  and  without  an  xy-term,  gives  a  parabola 
for  its  graph. 

415.  Suppose  a  line  to  start  from  the  position  marked 
y  =  2x—l,  moving  across  the  parabola  parallel  to  the  starting 
position  to  the  line  y  =  2x.  In  every  position  there  would  be 
two  crossing-points  until  the  position  y  =  2x  is  reached.  At 
this  position  the  two  crossing-points  blend  into  one,  the  line 
becoming  tangent  to  the  parabola. 

Beyond  the  position  y  —  2x  there  would  be  no  crossing- 
point  of  the  line  and  the  parabola. 

Starting  from  the  line  y  =  2x—l  and  moving  parallel  to 
itself  toward  the  right,  there  would  always  be  two  crossing- 
points.  Recalling  that  every  crossing-point  gives  a  value  of 
x  and  of  y,  we  observe  that: 

/.  There  are  in  general  two  solutions  of  a  system  made  up 
of  a  parabolic  and  a  linear  equation. 

II.  When  the  line  is  tangent  to  the  parabola  there  is  but  one 
solution,  or  since  the  tioo  crossing-points  coalesce,  we  may  say 
two  equal  solutions. 


SIMULTANEOUS  SYSTEMS  307 

III.  For  an  equation  representing  a  Ime  beyond  the  tangent 
position  there  is  no  real  solution.  Algebra  shows  that  there  are 
two  solutions  eve7i  here,  hut  that  they  are  imaginary. 

416.  The  solution  just  given  is  the  graphical  solution 
of  the  system.  We  now  give  the  algebraic  solution  of  the 
same  system. 

Writing  the  equations  thus : 

x^-4x-\-y+l=0  (1) 

y  =  2x-l  (2) 

?ubstitute  the  value  of  y  from  (2)  in  (1),  simplify,  and  find: 

Whence,  x  =  0,  and  +2 

Substituting  these  values  of  x  in  (2) ,  we  find : 
y=  —1,  and  -h3 

The  solutions  are  the  number  pairs : 

x  =  0,  x=-\-2, 
and      ?/=-l,2/  =  +3 

These  values  agree  with  those  of  the  graphical  solution. 


Exercise  181 

Solve  the  following  systems  algebraically: 

fx^-\-Sx-y  =  lS  /2.T2-6x-f2/  =  8 

\          y  —  2  =  2x  '  \          y  —  4x=—4: 

y'--2y-\-x  =  5  fy^-oy-\-3x  =  Q 

x-2y  =  S  '  \        2y-^x  =  4 

/a;2-?/  =  5  fSx^-9x-y  =  2 

\Zx~y=-5  \          3x-y  =  2 


308  ELEMENTARY   ALGEBRA 

417.  Solve  next  the  system  x^-\-i/  =  25    i/  =  =fc  \/25-xH]) 

or  U^+y'-  =  25,  or  ?/  =  ±  V^S  -  x^       (1) 

.(^  x-y=l,ory  =  x-l  (2) 

Graphing  (1)  y=  =t  \/25  —  x'^  using 

a;=  +6,  +5,  +4,  +3,     +2,     +1,  0,     -1,      -2,    -3,    -4,  -5,  -6, 

etc.,  and  calculating  y  from  ?/  =  ±  -\/25  —  x*.  find 

y=imag.      0,  =±=3,  ±4,  ±4.0,  ±4.9,  ±5,  ±4.9,  ±  4.6,  ±4,  ±3,     0,  imog., 

etc. 

Graphing  these  pairs,  laying 
off  the  values  with  double  sign 
both  upward  and  downward, 
obtain  the  circle  of  the  figure. 

Graphing  now  the  line  y  = 
x—l,  obtain  the  straight  Hne 
of  the  figure. 

The  crossing-points  give  the 
following  solutions : 

x=+4,  x=-3, 
t/=+3,  y=-4. 

This  is  the  graphical  solution. 

Suppose  a  line  should  start  from  the  position  x  —  y  =  l  and 
move  upward  across  the  circle,  keeping  parallel  to  x  —  y  =  \j 
through  the  positions x  —  y  =  0,x  —  y=—Z,iox  —  y=—iS \/2, 
or  downward  through  the  position,  x  —  y  =  ^  to  x  —  y  =  b\^2. 
In  every  position  the  line  gives  two  crossing-points  with  the 
circle,  until  the  tangent  positions  are  reached,  where  the 
two  crossing-points  become  one  point  of  contact. 

For  a  line  beyond  the  tangent  positions  the  system  would 
give  two  imaginary  solutions.  For  the  tangent  positions 
of  the  line  we  might  again  say  there  are  two  equal  solutions. 

For  the  upper  tangent-point  x=  —■^\/2,  y=-j-^\/2  and 
for  the  lower  tangont-point,  x=  -\-^\^2,  y=  — ■§-\/2. 


T     l¥^ 

z     / 

X      M^ 

^^z     zz 

r^^' 

yK.  // 

-j^    7 

-.^t   ^ 

*^1Z 

,7    LjZ    o 

T7     7 

/  '^^  \y     / 

z    zz 

*  '^jA- 

^zv:z^_. 

^-¥^ 

7     M/ 

7i\2'^ 

/TH^y 

Qf^ 

zz    z 

uz 

zz     z 

VL 

y 


Scale 
=s     horizontal  ^pace 
^     vertical    space 


SIMULTANEOUS   SYSTEMS 


309 


418.  The  algebraic  solution  consists  in  substituting  the 
value  of  y  from  (2)  in  (1),  obtaining: 


Or, 
or, 

Whence,  x  =  J±-|-= +4,  or  -3, 

Substituting  these  values  of  x  in  (2)  then  gives: 

2/= +3,  or  -4. 
These  solutions  agree  with  those  of  the  graphical  metho'd. 

Exercise  182 
Solve  the  following  systems  algebraically : 


4. 


7. 


2. 


5. 


8. 


a;2  4-|/2^58 
x  —  4iy=  —5 
a;2+i/2  =  29 
2a;-6?/  =  2 

9a:-2/  =  2 


3. 


6. 


9. 


6a;-5^=-26 

r^2_^^2=74 

\2a:+y  =  19 


x2+7/2  =  29 

a;2+7/2  =  52 
3a;-4^j  =  2 

419.  Solve  the  system : 

r       4a;-5i/  =  20      (1),  or  j/  =  |(x-5) 
^    \l6x2+25i/2  =  400    (2),  ori/  =  f\/25-x2. 

Calculate  the  ^/-values  for   (2)   from  these  aj-values: 

a:=+6, +5,     +4,     +3,     +2,     +1,     0,     -1,    -2,     -3,    -4,-5, 
ij=imag.    0,  ±2.4,±3.2, 

Y 


6, 


3.6, 


.X 


ij 

^ 

r^ 

,^ 

^ 

V 

I 

f 

^ 

V 

\ 

/ 

\ 

,, 

] 

0 

/ 

A 

\ 

&Y 

•- 

\ 

\d 

> 

'U 

^ 

■>~< 

\^ 

k 

rvn-\  - 

Y' 

Scal6 
1  ™  1    horizontal    space 
1  =  1     vertical   dpace 


3.9,  ±4,  ±3.9,  ±3.6,  =^3.2,  ±2.4,  0,imag. 
and  graph  (1)  using  x=+S, 
2/=— 1.6,  and  x  =  0,  ^=—4, 
obtaining  the  figure  here. 

The  number-pairs  of  the 
crossing-points  are  x=+5,  and 
0,  and  i/  =  0  and  —4,  which  are 
the  solutions.  Show  from  the 
figure  that  —  x=-f5  goes  with 
^  =  0,   and  x  =  0   with   y=—4. 


310  ELEMENTARY   ALGEBRA 

420.  The  algebraic  solution  is  obtained  by  substituting 
the  value  of  y  from  (1)  in  (2),  obtaining 

16x2+25[t(x-5)P  =  400 
Reducing,  32^2  -  160a:+400  =  400 
Or,  a:^  —  5x  =  0 

Whence,  x  =  0,  and  +5,  and  from  (1)  i/=  —  4,  and  0. 

The  graph  shows  that  the  0-value  of  x  must  be  paired  with 
the  —  4-value  of  y,  and  that  the  +5  and  0  also  belong  to- 
gether. 

The  graph  for  the  equation  16x^+251/2  =  400,  is  an  ellipse. 

Moving  this  line  4a:  — 5i/  =  20  parallel  to  itself  across  the 
ellipse  shows  there  are  always  two  crossing-points,  and  hence 
two  pairs  of  values  of  x  and  y,  save  for  the  tangent  positions, 
where  there  would  be  only  one  pair  or,  as  we  prefer  to  say, 
two  equal  pairs. 

An  algebraic  solution  would  show  that  when  the  line  does 
not  touch  the  ellipse  there  would  be  two  imaginary  values 
of  X  and  y. 

421.  A  quadratic  equation  with  no  xy-term.  but  containing 
the  square-terms  of  both  variables,  the  coefficients  of  these 
terms  being  of  the  same  sign,  gives  a  graph  that  is 'an  ellipse. 

Exercise  183 

Solve  the  following  systems  algebraically: 

f   2x-Zy  =  0  (     x-3y  =  2  (  Sx-5y  =  S 

\4a;2+92/2  =  36  \4x^+9y''  =  SQ  \x'+25y^  =  25 

(  7x-4y  =  10  (5x-.3y  =  S  (l0x-Sy  =  5     , 

\x^+lQy^  =  m  \9x2+y2  =  9  \49a;2+^2  =  49 

422.  Solve  the  system: 

-1/2=16  '  (1),  or2/==*=V^^^^ 

-2/  =  2  {2),ory  =  x-2 


SIMULTANEOUS  SYSTEMS 


311 


In  equation  (1)  for  all  values  of  x  between  —4  and  +4  the 
values  of  y  are  imaginary.  Calculate  y  for  the  given  .re- 
values, find: 

x=+10,       +8,    +5,   +4,     -4,     -5,         -8,      -10,  etc. 
^=±9.2,   ±6.9,   ±3,       0,        0,      ±3,      ±6.9,      =^9.2,  etc. 

Plotting  these  points,  drawing  the  graph,  and  graphing 
equation  (2)  for  x  =  0,  y=—2,  and  x= —4,  y=—Q,  obtain 
the  picture  of  the  figure  shown. 

The  graph  of  equation  (1)  is  a  hyperbola.  It  has  two 
disconnected  parts,  or  branches.     There  is  but  one  crossing- 


1 

9v 

.-i- 

^ 

.^^ 

s 

.^^ 

Nk 

^^ 

Y 

"  5c 

o 

X^' 

J 

n         -^f' 

^ 

y^    "v* 

^v 

^ 

>v 

^^     '^ 

\^ 

^   ^q^ 

s 

ifc  -=^ 

^ 

^^ 

± 

Scale 
]   :=  1  horizontal  space 
2=1    vertical    space 


point  of  the  line  and  curve.     The  figure  shows  why.     The 
graph  shows  the  x-  and  ^/-values  for  this  crossing-point  to  be 

x=  -}-5,  and  y=  +3. 

423.  The  algebraic  solution  gives  by  substituting  the  value 

of  2/ from  (2)  in  (1)    x^-{x-2Y=\^ 

Reducing,  we  find,  4a:  =  +20, 

or,  x=+5. 

This  value  of  x,  substituted  in  equation  (2),  gives 

2/=+3. 

These  values  of  x  and  y  agree  with  the  graphical  solution. 


312  ELEMENTARY   ALGEBRA 

424.  A  quadratic  equation  having  both  x--  and  y^-terms 
with  opposite  signs,  no  xy-term  being  present,  always 
gives  a  hyperbola  for  its  graph. 

Could  the  straight  line  be  turned  around  so  that  it  would 
cut  both  branches  of  the  hyperbola?  How  many  values  of  x 
and  of  y  would  thece  be? 

Exercise  184 

Solve  the  following  systems  algebraically: 

(x^-y^  =  7  (x^-y^  =  lS  (x'~-y^  =  ^5 

'  \  x-y=l  *  \  x-y=l  '  \  x-y  =  5 

*•  ^    x-y  =  3  \  x-y  =  5  \x-3y  =  Q 


425.  Solve  the  system : 

xy  =  12,  or  y  =  ^  (1) 


12 


X 

y-x  =  l,     or  t/  =  x+l  (2) 

In  equation   (1)   calculate  y  for  the  following  assumed 
values  of  x, 

a;=+12,-f6,+4,-f3,+2,    +1,-   1,-2,-3,-4,-6,-12 
y=    +l,+2,+3,+4,+6,+12, -12, -6,-4,-3,-2,,    -1 

Plot  these  points,  and  draw  the  graph,  obtaining  a  curve 
for  xy  =  12.     Show  both  branches  of  the  curve. 

Both  branches  together  are  spoken  of  as  a  single  curve, 
the  hyperbola. 

Graphing  equation  (2)  on  the  same  axes,  using  the  follow- 
ing points, 

a:=+3,      0,   -1,   -4 
y=+4,  +1,       0,   -3, 

the  straight  line  graph  for  y  =  x-\-l  is  obtained. 


SIMULTANEOUS   SYSTEMS  313 

426.  The  roots  are  the  x-  and  i/-values  of  the  crossing- 
points  of  the  two  graphs,  viz. : 

x=+3  and    (x=  —4 
y=+4  12/= -3 

Check:  Substitute  the  number-pair  (+3,  +4)  in  equations  (1)  and 
(2)  thus, 

in(l),  +4  =  4^  and  in  (2),  +4= +3  +  1. 

+  3 

Then,  substitute  the  other  number-pair  (  —  4,  —3)  in  (1)  and  (2)  thus, 

in  (1),  -3  =  —  .  and  in  (2),  -3=  -4  +  1. 
—  4 

Hence  the  pairs  (+3,  +4)  and  (  —  4,  —3)  are  the  root- 
pairs  of  the  given  system. 

427.  Equations  Uke  those  of  the  system  of  §  425  have  a 
hyperbola  and  a  straight-line  for  graphs.  The  solutions,  or 
roots,  of  the  system  are  the  x-  and  ^/-values  of  all  the  crossing- 
points.  Such  graphs  in  general  have  two  crossing-points, 
and  hence,  two  a;-values  and  two  ^/-values,  and  these  x-  and 
i/-values  must  be  so  paired  that  the  two  numbers  of  a  pair  shall 
belong  to  the  same  crossing-point  of  the  graphs. 

428.  The  algebraic  solution  of  the  system  of  §  425, 

xy  =  l2  (1) 

_y-x  =  l  (2) 

is  as  follows: 
From  (2)  we  have 

y^x-1  (3) 

Substitute  —  (3)  in  (1),  obtaining 

x(x-l)  =  12,  or 
a;2-x  =  12 
Whence,  x=+3,  or  —4 

Substituting  these  values  of  x  in  (1),  we  obtain 

x=+4,  or  -3 
These  are  the  values  given  by  the  graphical  solution  of 
§425. 


314  ELEMENTARY  ALGEBRA 

Exercise  185 

xy  =  12  2     f     xy  =  36         ^     (       xy  =  20 

y-x  =  4:  '\x—y=—5         '\x-4:y  =  2 

3xy  =  21  g    I  5xy  =  150       ^     (     7xy  =  9S 

x  —  8y=—l  '  \x-y=—l         '  \x-5y=-3 

429.  The  main  use  of  the  graphical  solution  of  equations 
to  pupils  is  to  enable  them  to  see  the  meaning  of  solutions, 
and  to  understand  why  roots  are  paired  in  a  certain  way. 

For  practical  work  of  solving  equations  the  algebraic 
solution,  as  given  in  §§  416,  418,  420,  423  and  428,  should 
always  be  used.  In  the  exercises  that  follow  the  algebraic 
method  is  to  be  employed. 


Exercise  186 
Solve  the  following  systems  and  pair  the  roots  properly 


3. 


X  -y  =  3  '   [  x-i-y  =  10 

y-x^-\-x=l  ^    jx^+y^  =  20 


x  =  y—4:  .  [  x-\-y  =  Q 

x2+i/2  =  26  ^    fx^+y^  =  7S 

6. 


x-y  =  6  [y-2x  =  lS 

3x2+81/2=147  ^    /x2+2i/2  =  89 

8> 


x  —  y  =  2  [     a:+^  =  ll 

(5x'-\-y'  =  4:5  I     xy  =  10 

•  \  x+2y  =  12  ^^'  \y-x  =  S 

11.  <     ^^=^^  12    ^     ^'^^^^ 


x-\-y  =  9  [x  —  y  =  2 

f2?/  =  10  ^^     (Sx^-y  =  7 

x—y  =  7  '  \  y  —  5x  =  5 


SIMULTANEOUS   SYSTEMS  315 

16.  <       ^'=12-2/  16./"+"  =  ^ 


^^     'm-n  =  S  .-    /a;2+a;?/+i/2  =  61 


mn=18  \  x+7/  =  9 


a+c  =  14  \         y  =  ll—x 

21.  <  „  22. 


?yi-n  =  3  [    3i/-2a;=l 

fm?-\-n?+mn  =  S9  (xy+y^  =  40 

^^'  \  m-n  =  3  ,  \i/-3a:=-4 


a2-3i/=l3  \       mx  =  85 

^      V4-62-a-6=18  „„    /a^2_^23=65 

27.   <  ,  ,  r  28. 


a+6=-5  ]t/-2a:=-14 


2^      a6+a2  =  40  ,         30.  J  ^^"^—^ 


_  15 
~    4 

6_3a=-4  *""  \   x-i/  =  f 

;c2-5d2  =  76  ,„     f2m-3n  =  9 

31.  <  .        .  ,     ^^  32. 


4c-5rf  =  29  \         mn  =  Q 

gg     (Sxy-hx'-2y'=^52  ^^    j  3m-2n  =  2S 


\  2x+Sy  =  SQ  '  \m''-27nn  =  45 


cd  =  57  '         '   \    y-z=lQ 

37.  ;^^-^^  =  16  33^   /     xy  =  4 


n+2m=13  {x-y  =  0 

;4m2-9n2=19  ,^    /p2+4^  =  76 

39.   <    „     .  ^         .^  40.   ^"^         ^ 


3n+2m=19  '   \Sp-q  =  21 

7nn  =  30  '  \  a;-3t/=-5 


,m2+n2-m-n  =  50  ^„    j3a;2-2/2  =  275 

41.   N  «,x  42. 


316  ELEMENTARY  ALGEBRA 

430.  special  Methods.  Some  systems  may  be  conven- 
iently solved  by  special  methods  as  well  as  by  substitution. 

431.  One  of  these  special  methods  is  to  divide  the  given 
equations,  member  by  member,  obtaining  a  derived  equation 
which,  with  one  of  the  given  equations,  furnishes  a  system  of 
equations  equivalent  to  the  given  system,  and  then  to  solve 
the  derived  system. 

(a)  Observe  carefully  the  following  solution  of  the  system : 
'2-1/2  =  33  (1) 
x-i-y  =  n  (2) 

Dividing  (1)  by  (2),     x-y^S  (3) 

The  system  consisting  of"  (2)  and  (3)  is  simpler  than  the 

given  system  and  the  simpler  system  gives  x  =  7  and  t/  =  4. 
These  are  all  the  roots,  for  (1)  represents  a  hyperbola  and 

(2)  a  straight  line,  and  they  cross  in  only  one  point. 

(b)  Solve  the  system : 

/36m?-p2  =  819  (1) 

\     67n-p=-39  (2) 

Dividing  (1)  by  (2),      Q7n-{-p=-21  (3) 

The  system  (2)  and  (3)  is  equivalent  to  the  given  system 

and  its  roots  are : 

m=  —  5  and  p=  +9 

Exercise  187 

Solve  the  following  systems,  first  dividing  when  possible 
and  pairing  results  properly: 

'93.2  _  4^2  ^  308  /m2  -  n2  =  64 


^'   '    Sx-2y  =  14:  "   1    m+n  =  16 


3.  <  /  4.  < 

3p  — =  8 
Q 


m    m 

-1~I2  =  ^^ 
or    If 

m    m 
— -=n 
a      0 


SIMULTANEOUS  SYSTEMS 


317 


5. 


7. 


3a;2=16-25?/2 


7?i  =  f -|-9p 


6.  < 


8.   < 


1 

1     ^ 

?" 

--2  =  5 

1 

1 

-H- 

-  =  5 

.^ 

2/ 

f  ^ 

32 

=  6 

Ri' 

i22^ 

1 

2 

■+-B-  = 

1.5 

.^1 

/^2 

432.  Another  special  method,  or  device,  is  to  form  systems 
equivalent  to  the  given  system  by  so  combining  the  given 
equations  as  to  obtain  squares  in  both  members  and  then  to 
derive  simpler  systems  by  extracting  the  square  roots  of 
both  members. 

(a)  Observe  carefully  the  following  solution  of  the  system : 

x2+?/2+a:2/  =  52  (1) 

x+y  =  S-  (2) 

Subtracting  the  first  equation  from  the  square  of  the 

second,  xy  =  12  (3) 

The  system  made  up  of  (2)  and  (3)  is  equivalent  to  the 

system  (1)  and  (2). 

The  system  (2)  and  (3),  solved  as  in  §  428,  gives  x  =  6 
and  2  and  y  =  2  and  6. 
(6)  Solve  the  system: 

x'^-{-xy=10 
2/2-hxi/=15 

Adding  the  equations  we  obtain : 

{x-\-yy  =  25,  or  x-\-y=  ±5 
Subtracting  (1)  from  (2), 

y^  —  x'^  =  5 
Equation  (3)  is  really  two  equations,  viz. : 
x-\-y  =  5,  Siud  x  —  y= —5. 


(1) 

(2) 
(3) 
(4) 


318  ELEMENTARY   ALGEBRA 

The  given  system  is  then  equivalent  to  the  two  systems: 
^'-y'-=-^  and  J^'-f=-5 


x+y=-{-5  {  x-]-y=  -5 

Dividing  the  first  equations  by  the  second,  obtain: 

x  —  y=—l  and  x  —  y=-\-l 
Combining  these  with  the  second  equations  of  the  derived 
system,  we  have: 

x  =  2  and  y  =  S,  and  x=  —2  and  y=—d 

(c)  Solve  the  system: 

x2+2/2  =  40  (1) 

xy  =  l2  (2) 

Multiplying  (2)  by  2  and  adding  to  (1),  obtain: 

x2+2x?/+2/2  =  64 
Or,  x+y==^S  (3) 

Subtracting  2xy  =  24:  from  (1) 

x^-2xy-{-y^=m, 
Or,  x-y=  ±4  (4) 

Now  from  (3)  and  (4)  we  form  the  four  systems  which  are 
together  equivalent  to  the  given  system,  viz. : 

J      x+y=+S  jj    (x+y=-\-S 


x  —  y=-}-4:  [x  —  y=—4: 

III.  ^+^=7^  IV.  l^+y-'l  . 

[x-y=-]r4:  [x  —  y=—4: 

System  I  gives  x  =  Q,  y  =  2,  II,  gives  x  =  2,  y  =  Q,  III,  gives 
x=  —2,  y=  —6,  and  IV  gives  x=  —6,  y=  —2.  Hence,  the 
solutions  of  the  given  system  are : 

x=+6,  +2,  -2,  and  -6, 
y=-\-2,  +6,  -6,  and  -2. 

The  system  (1)  and  (2)  are  both  quadratic  equations,  so  that  this 
problem  lies  a  little  beyond  the  limits  set  for  this  book.  But  the  method 
in  most  of  its  parts  is  so  like  that  for  systems  made  up  of  one  quadratic 


SIMULTANEOUS   SYSTEMS  319 

and  one  linear  as  to  bring  it  within  the  pupil's  comprehension.  The 
reason  there  are  so  many  solutions  lies  in  the  fact  that  the  graph  of 
(1)  is  a  circle  and  of  (2)  a  hyperbola,  since  a  circle  and  a  hyperbola, 
in  general,  cross  each  other  in  four  points. 

433.  In  the  following  list  of  exercises  we  shall  include  a 
few  systems  in  two  quadratics  of  the  type  of  the  last. 


Exercise  188 
Solve  the  following  systems  of  equations: 

Qxy=lS  '  \  rs=  12 

^'  \         Smn=m  ^  *•   I  11  =  3 

x''-xy  =  22  (x''+4:xy+3Qif  =  224 

^xy-y^=lS   ■  \  I2xy  =  m 


Exercise  189 

1.  The  sum  of  two  numbers  is  7  (or  a),  and  the  sum  of 
their  squares  is  21  (or  b) .     What  are  the  numbers  ? 

2.  Find  two  numbers  the  difference  of  whose  squares  is 
33  (or  m),  and  the  product  of  whose  squares  is  784  (or  n). 

3.  The  combined  area  of  two  square  fields  is  8|  acres, 
and  the  sum  of  their  perimeters  is  200  rods.  What  is  the 
area  of  each  field  ? 

4.  The  sum  of  the  squares  of  two  numbers  is  91  (or  p),'and 
the  difference  of  the  numbers  is  5  (or  q).     Find  the  numbers. 

5.  The  difference  of  two  numbers  is  28,  and  half  their 
product  is  equal  to  the  cube  of  the  smaller  number.  What 
are  the  numbers  ? 


320  ELEMENTARY   ALGEBRA 

6.  The  area  of  the  ceiUng  of  a  hall  is  700  square  feet, 
and  its  length  is  six  feet  less  than  four  times  the  width. 
Find  the  dimensions. 

7.  The  sum  of  two  numbers  is  13  (or  s),  and  their  product 
is  210  (or  p).     Find  the  numbers. 

8.  If  the  dimensions  of  a  rectangle  were  each  increased 
1  foot,  the  area  would  be  99  square  feet;  if  they  were  each 
diminished  1  foot,  the  area  would  be  63  square  feet.  What 
are  the  dimensions? 

9.  A  number  is  expressed  by  two  figures  the  sum  of 
which  is  14,  and  the  sum  of  the  squares  of  the  digits  exceeds 
the  number  by  11.     Find  the  number. 

10.  The  combined  area  of  two  adjoining  square  fields  is 
900  square  rods,  and  it  requires  150  rods  of  fence  to  inclose 
them.  If  they  are  so  situated  as  to  require  the  least  amount 
of  fence,  what  is  the  dimension  of  each  ? 

11.  The  area  of  a  rectangle  is  192  square  inches,  and  its 
diagonal  is  20  inches.     Find  the  dimensions. 

12.  A  rectangular  field  contains  270  square  rods.  If  it 
were  two  rods  longer  and  one  rod  wider,  it  would  contain 
50  square  rods  more.     Find  the  dimensions  of  the  field. 

13.  A  farmer  bought  12  sheep  and  4  calves  for  $60.  At 
the  same  prices,  he  could  buy  3  more  sheep  for  $24  than 
calves  for  $30.     Find  the  price  of  each. 

14.  The  perimeter  of  a  rectangular  piece  of  ground  is 
200  rods,  and  its  area  is  15  acres.  Find  the  dimensions  of 
the  field. 

15.  The  hypotenuse  of  a  right  triangle  is  30  feet, 
and  its  area  is  216  square  feet.  Find  the  length  of  the  other 
two  sides. 


SIMULTANEOUS   SYSTEMS  321 

16.  The  sum  of  the  squares  of  two  numbers  is  74,  and 
the  difference  of  their  squares  is  24.     What  are  the  numbers  ? 

17.  A  merchant  bought  two  kinds  of  silk,  paying  $63 
for  each  piece,  and  buying  8  yards  more  of  one  kind  than 
the  other.  The  difference  in  price  was  50  cents  a  yard. 
How  many  yards  of  each  kind  did  he  buy  ? 

18.  A  rectangular  piece  of  paper  contains  1350  square 
inches;  but  if  the  dimensions  were  each  5  inches  less,  it 
would  contain  1000  square  inches.     Find  the  dimensions. 

19.  If  the  sum  of  two  numbers  is  added  to  their  product, 
the  result  is  31;  and  the  sum  of  their  squares  exceeds  their 
sum  by  48.     What  are  the  numbers  ? 

20.  A  man  bought  sheep  for  $136.  He  kept  22  of  them, 
and  sold  the  remainder  at  a  profit  of  $1  a  head,  receiving  for 
them  $2  more  than  they  all  cost.  At  what  price  per  head 
did  he  buy  them  ? 

21.  The  square  of  the  sum  of  two  numbers  exceeds  6  times 
the  sum  of  the  numbers  by  16.  The  difference  of  the  num- 
bers is  2.     Find  the  numbers. 

22.  The  opposite  sides  of  a  parallelogram  are  equal.  One 
pair  of  opposite  sides  of  the  parallelogram  are  denoted  by 
m^—mn  and  19  — n^,  and  the  other  pair  are  denoted  by  7n 
and  n+3.     Find  m  and  n  and  the  length  of  the  sides. 


SUMMARY  OF  DEFINITIONS  FOR  REFERENCE  AND  REVIEW 

(Definitions  without  page  numbers  are  on  page  last  indicated.) 

CHAPTER  I 

The  factors  of  a  number  are  its  makers  by  multiplication.     (Page  8.) 
An  equation  is  an  expression  of  equality  between  two  equal  numbers. 
(Page  11.) 

The  value  of  any  letter  in  a  number  expression  is  the  number  or 
numbers  it  represents.     (Page  12.) 

An  unknown  number  is  a  letter  whose  value  in  an  equation  is  to 
be  found.     (Page  13.) 

Solving  an  equation  is  finding  the  value  of  the  unknown  number,  or 
numbers  in  it. 

An  axiom  is  a  statement  so  evidently  true  that  it  may  be  accepted 
without  proof. 

In  problem-solving  the  notation  is  the  representation  in  algebraic 
symbols  of  the  unknown  numbers  of  the  problem.     (Page  15.) 

The  statement  is  the  expression  of  the  conditions  of  the  problem 
in  one  or  more  equations. 

CHAPTER  II 

Directed  numbers  or  signed  numbers  are  numbers  whose  units  are 
positive  or  negative.     (Page  21.) 

The  absolute  value  of  a  number  is  the  number  of  units  in  it,  regard- 
less of  sign.     (Page  22.) 

The  +  and  —  signs  may  denote  either  operations  or  opposing  qual- 
ities of  numbers.     (Page  23.) 

Algebraic  notation  is  a  method  of  expressing .  numbers  by  figures 
and  letters.     (Page  24.) 

An  algebraic  expression  is  the  representation  of  any  number  in 
algebraic  notation. 

A  term  is  a  number  expression  whose  parts  are  not  separated  by 
the  +  or  —  sign. 

A  monomial  is  an  expression  of  one  term.     (Page  25.) 

A  polynomial  is  an  expression  of  two  or  more  terms. 

322 


SUMMARY  OF   DEFINITIONS  323 

A  binomial  is  a  polynomial  of  two  terms. 

A  trinomial  is  a  polynomial  of  three  terms. 

A  coefficient  of  a  term  is  any  factor  of  the  term  which  shows  how 
many  times  the  other  factor  is  taken  as  an  addend. 

Similar  terms  are  terms  which  do  not  differ,  or  which  differ  only  in 
their  numerical  factors. 

Dissimilar  terms  are  terms  that  are  not  similar. 

Partly  similar  terms  are  terms  that  have  a  common  factor. 

The  value  of  an  algebraic  expression  is  the  number  it  represents 
when  some  particular  value  is  assigned  to  each  letter  in  the  expression. 
(Page  26.) 

CHAPTER  III 

Addition  is  the  process  of  uniting  two  or  more  numbers  into  one 
number.     (Page  27.) 

The  addends  are  the  numbers  to  be  added. 

The  sum  is  the  number  obtained  by  addition. 

The  fundamental  laws  of  addition  are  the  law  of  order,  (the  com- 
mutative law),  and  the  law  of  grouping  (the  associative  law).     (Page  29.) 

The  law  of  order  states  that  numbers  may  be  added  in  any  order. 

The  law  of  grouping  states  that  addends  may  be  grouped  in  any  way. 

CHAPTER   IV 

Subtraction  is  the  process  of  finding  one  of  two  numbers  when  their 
sum  and  the  other  number  are  known.     (Page  35.) 

The  minuend  is  the  number  that  represents  the  sum. 

The  subtrahend  is  the  given  addend. 

The  difference  or  remainder  is  the  number  which  added  to  the  sub- 
trahend gives  the  minuend. 

The  symbols  of  aggregation  are  the  parenthesis  (  ),  the  brace  [  } ,  the 
bracket  [  ],  and  the  vinculum  ~".     (Page  42.) 

CHAPTER  V 

Algebraic  number  and  function  have  the  same  meaning.     (Page  50.) 
The  independent  number  is  the  number  on  which  the  function 
depends. 


324  ELEMENTARY  ALGEBRA 

A  function  is  a  number  that  depends  on  some  other  number  for  its 
value.     (Page  5L) 

An  algebraic  function  is  a  number  whose  dependence  on  another 
number  is  expressed  in  algebraic  symbols. 

CHAPTER  VI 

Equations  are  of  two  kinds,  identities  and  conditional  equations. 
(Page  60.) 

An  identity  is  an  equation  with  like  members,  or  members  which 
may  be  reduced  to  the  same  form. 

Substitution  is  putting  a  number  symbol  into  a  number  expression 
in  place  of  another  which  has  the  same  value. 

An  equation  is  satisfied  by  any  number  which,  when  substituted 
for  the  unknown  number,  reduces  the  equation  to  an  identity. 

A  conditional  equation  is  an  equation  that  can  be  satisfied  by  only 
one  or  by  a  definite  number  of  values  of  the  letters  in  it.     (Page  6L) 

A  root  of  an  equation  is  any  value  of  the  unknown  number  that 
satisfies  the  equation. 

Transposition  is  the  process  of  changing  a  term  from  one  member  of 
an  equation  to  the  other,  by  adding  or  subtracting  the  same  number 
in  both  members.     (Page  62.) 

CHAPTER  VII 

Graphing  means  representing  number-pairs,  related  sets  of  numbers, 
and  number  laws  by  pictures  and  diagrams.     (Page  74.) 

A  linear  equation  is  an  equation  in  two  unknowns  both  with  exponent 
1.     (Page  8L) 

The  graphical  solution  of  two  linear  equations  is  the  point  of  inter- 
section of  the  graphs  of  the  equations.     (Page  82.) 

Simultaneous  equations  are  equations  that  can  be  satisfied  by  the 
same  values  of  x  and  y. 

A  system  of  equations  is  two  or  more  equations  considered  together. 
(Pages  82  and  86.) 

Non-simultaneous  or  inconsistent  equations  are  equations  which 
cannot  be  satisfied  by  any  values  of  the  unknowns.     (Page  83.) 

Dependent  equations  are  equations  in  which  one  or  more  can  be 
derived  from  another  or  others  by  some  simple  arithmetical  operation. 
(Page  84.) 


SUMMARY  OF   DEFINITIONS  325 

CHAPTER  VIII 

A  determinate  equation  is  an  equation  which  has  one  root,  or  a 
limited  number  of  roots.     (Page  85.) 

An  indeterminate  equation  is  an  equation  which  has  an  unlimited 
number  of  roots. 

Independent  equations  are  equations  which  cannot  be  derived 
one  from  another  by  a  simple  arithmetical  operation.     (Page  86.) 

A  set  of  roots  of  a  system  of  equations  means  the  values  of  the  un- 
known numbers  of  the  system. 

Elimination  is  a  process  of  deriving  a  single  equation  in  one  unknown 
from  a  system  of  two  or  more  simultaneous  equations  in  two  or  more 
unknowns. 

CHAPTER  IX 

Multiplication  is  the  process  of  taking  one  number  as  an  addend 
a  certain  number  of  times.     (Page  91.) 

The  multiplicand  is  the  number  taken  as  an  addend. 

The  multiplier  is  the  number  denoting  how  many  times  the  multi- 
plicand is  taken. 

The  product  is  the  result  of  the  multiphc.ation. 

A  negative  multiplier  means  that  the  product  is  of  the  opposite 
quality  from  what  it  would  be  if  the  multiplier  were  positive. 

An  exponent  is  a  symbol  of  number  written  at  the  right  and  a  httle 
above  another  symbol  of  number  to  show  how  many  times  the  latter 
is  taken  as  a  factor.     (Page  92.) 

The  three  fundamental  laws  of  multiplication  are  the  law  of  order 
(commutative  law),  the  law  of  grouping  (associative  law),  and  the 
distributive  law.     (Page  94.) 

The  law  of  order  is:  The  prod^ict  of  several  numbers  is  the  same  in 
whatever  order  they  are  u^ed. 

The  law  of  grouping  is:  The  product  of  several  numbers  is  the  same 
in  whatever  manner  they  are  grouped. 

The  distributive  law  is:  The  product  of  a  polynomial  and  a  monomial 
is  the  algebraic  su7n  of  the  products  obtained  by  multiplying  each  term 
of  the  polynomial  by  the  monomial.     (Page  95.) 

A  power  is  the  product  obtained  by  taking  a  number  any  number 
of  times  as  a  factor. 


326  ELEMENTARY  ALGEBRA 

A  polynomial  is  arranged  when  the  exponents  of  some  letter  increase 
or  decrease  with  each  succeeding  term.     (Page  97.) 

CHAPTER  X 

The  degree  of  a  term  is  indicated  by  the  sum  of  the  exponents  of 
the  literal  factors.     (Page  100.) 

The  degree  of  an  equation  in  one  unknown  is  the  degree  of  the 
highest  power  of  the  imknown  number. 

A  simple  equation,  or  linear  equation,  is  an  equation  which,  when 
cleared  and  simpUfied,  is  of  the  first  degree. 

Checking  or  verifying  a  root  of  an  equation  is  the  process  of  proving 
that  the  root  satisfies  the  equation. 

CHAPTER   XI 

Division  is  the  process  of  finding  one  of  two  numbers  when  their 
product  and  the  other  number  are  known.     (Page  107.) 

The  dividend  is  the  number  to  be  divided  and  represents  the  product 
of  the  two  numbers. 

The  divisor  is  the  number  by  which  we  divide  and  represents  one 
factor  of  the  dividend. 

The  quotient  is  the  result  of  division. 

Any  number  with  a  zero-exponent  equals  1.     (Page  108.) 

CHAPTER  XIII 

A  general  number  is  a  letter  or  other  number  symbol  that  may  repre- 
sent any  number.     (Page  123.) 

A  formula  is  an  expression  of  a  general  principle,  or  rule,  in  general 
number  symbols  and  in  the  form  of  an  equality.     (Page  124.) 

To  solve  a  formula  completely  is  to  find  the  value  of  each  general 
number  in  terms  of  the  others.     (Page  125.) 

CHAPTER  XIV 

A  root  of  a  number  is  one  of  its  equal  factors.     (Page  139.) 

The  square  root  of  a  number  is  one  of  the  two  equal  factors  whose 

product  is  the  number.     (Page  140.) 

The  cube  root  of  a  number  is  one  of  the  three  equal  factors  whose 

product  is  the  number. 


SUMMARY  OF   DEFINITIONS  327 

CHAPTER  XVI 

A  common  divisor,  or  common  factor,  of  two  or  more  numbers  is 
an  exact  divisor  of  eacii  of  them.     (Page  172.) 

The  highest  common  factor  (h.c.f.)  of  two  or  more  numbers  is  the 
product  of  all  their  common  factors. 

A  multiple  of  a  number  is  a  number  that  is  exactly  divisible  by  it. 
(Page  175.) 

A  common  multiple  of  two  or  more  numbers  is  a  number  that  is 
exactly  divisible  by  each  of  them. 

The  lowest  common  multiple  (l.c.m.)  of  two  or  more  numbers  is 
the  product  of  all  their  different  factors. 

CHAPTER  XVII 

An  algebraic  fraction  is  the  indicated  division  in  fractional  form 
of  one  number  by  another.     (Page  179.) 

The  numerator  is  the  number  above  the  line. 

The  denominator  is  the  number  below  the  hne. 

The  terms  of  a  fraction  are  the  numerator  and  denominator  together. 

An  integer,  or  integral  number,  is  a  number  no  part  of  which  is  a 
fraction. 

The  sign  of  a  fraction  is  the  sign  written  before  the  line  that  separates 
the  terms.     (Page  180.) 

Reduction  of  fractions  is  the  process  of  changing  their  form  without 
changing  their  f  a/ wes.     (Page  181.) 

A  mixed  number  is  a  number  one  part  of  which  is  integral  and  the 
other  part  fractional.     (Page  184.) 

A  proper  fraction  is  a  fraction  which  cannot  be  reduced  to  a  whole 
or  a  mixed  number. 

An  improper  fraction  is  a  fraction  which  can  be  reduced  to  a  whole 
or  a  mixed  number. 

The  lowest  common  denominator  (Led.)  of  two  or  more  fractions 
is  the  l.c.m.  of  their  denominators.     (Page  187.) 

The  reciprocal  of  a  fraction  is  the  fraction  inverted.     (Page  193.) 

CHAPTER  XVIII 

A  literal  equation  is  an  equation  in  which  there  are  two  or  more 
general  numbers.     (Page  198.) 


328  ELEMENTARY  ALGEBRA 

A  general  problem  is  a  problem  all  of  the  numbers  in  which  are 
general  numbers,     (Page  207.) 

CHAPTER  XX 

The  ratio  of  one  number  to  another  is  the  quotient  of  the  first 
number  divided  by  the  second.     (Page  229.) 

The  antecedent  is  the  first  number  of  a  ratio,  and  the  consequent 
is  the  second  number. 

The  terms  of  a  ratio  are  the  antecedent  and  consequent. 

The  value  of  a  ratio  is  the  quotient  expressed  in  its  lowest  terms. 

A  ratio  of  greater  inequality  is  a  ratio  in  which  the  antecedent  is 
greater  than  the  consequent.     (Page  231.) 

A  ratio  of  less  inequality  is  a  ratio  in  which  the  antecedent  is  less 
than  the  consequent. 

A  proportion  is  an  equation  of  ratios.     (Page  232.) 

The  terms  of  a  proportion  are  the  terms  of  the  ratios. 

The  extremes  of  a  proportion  are  the  first  and  fourth  terms;  the 
means  are  the  second  and  third  terms. 

A  mean  proportional  is  the  second  of  three  numbers  which  form  a 
continued  proportion,  as  x  in  a: a:  =  x:b.     (Page  234.) 

A  third  proportional  is  the  third  of  three  numbers  that  form  a  con- 
tinued proportion. 

A  fourth  proportional  is  the  fourth  of  four  numbers  that  form  a 
proportion. 

A  variable  number,  or  a  variable,  is  a  number  which  in  a  given 
problem,  or  discussion,  may  have  different  values.     (Page  241.) 

A  constant  number,  or  a  constant,  is  a  number  that  is  not  a  variabl  \ 

One  variable  varies  as  another  if,  as  they  vary,  their  ratio  remains 
constant. 

CHAPTER  XXI 

Involution  is  the  process  of  raising  a  number  to  a  power  whose 
exponent  is  a  positive  integer.     (Page  244.) 

The  exponent  of  the  power  is  the  number  which  indicates  how  many 
times  the  number  (the  root  or  base)  is  taken  as  a  factor. 

The  base  of  a  power  is  the  number  which  is  raised  to  a  power. 

Evolution  is  the  process  of  finding  a  root  of  a  number.     (Page  250.) 


SUMMARY  OF  DEFINITIONS  329 

The  index  of  a  root  is  a  number  symbol  written  or  understood  in 
the  opening  of  the  sign  V     to  denote  what  root  is  intended. 

A  radical  is  any  root  of  a  number  indicated  by  the  radical  sign,  \/~> 
or  V    ,  or  by  a  fractional  exponent.     (Pages  250  and  264.) 

An  odd  root  is  a  root  whose  index  is  an  odd  number.     (Page  252.) 

An  even  root  is  a  root  whose  index  is  an  even  number. 

An  imaginary  number  is  an  indicated  even  root  of  a  negative  num- 
ber.    (Page  253.) 

A  real  number  is  a  number  that  does  not  involve  an  even  root  of  a 
negative  number. 

The  principal  root  of  a  number  is  the  real  root  which  has  the  same 
sign  as  the  number  itself.     (Page  254.) 

The  radicand  is  the  number  whose  indicated  root  is  to  be  found. 
(Page  264.) 

The  order,  or  degree,  of  a  radical  is  determined  by  the  index  of  the 
root.' 

A  rational  number  is  a  positive  or  negative  integer  or  a  fraction 
whose  terms  are  integers. 

An  irrational  number  is  a  number  which  cannot  be  expressed  wholly 
in  rational  form.     (Page  265.) 

A  surd  is  an  indicated  root  of  a  rational  number  which  cannot  be 
exactly  obtained. 

An  arithmetic  surd  is  a  surd  whose  radicand  is  an  arithmetical 
number. 

An  algebraic  surd  is  a  surd  whose  radicand  is  an  algebraic  expression. 

The  coefficient  of  a  radical  is  the  rational  factor  before  the  radical. 

A  pure  surd,  or  an  entire  surd,  is  a  surd  having  no  coefficient  ex- 
pressed. 

A  mixed  surd  is  a  surd  having  a  coefficient  expressed. 

A  quadratic  surd  is  a  surd  of  the  second  order. 

Similar  surds  are  surds  which  in  their  simplest  form  are  of  the  same 
degree  and  have  the  same  radicand.     (Page  271.) 

Rationalizing  a  surd  is  the  process  of  multiplying  the  surd  by  a 
number  that  gives  a  rational  product.     (Page  275.) 

The  rationalizing  factor  is  the  factor  by  which  a  surd  is  multiplied 
to  give  a  rational  product. 

A  binomial  surd  is  a  binomial  one  or  hath  of  whose  terms  are  surds. 


330  ELEMENTARY  ALGEBRA 

A  binomial  quadratic  surd  is  a  binomial  surd  whose  surd  term  or 
terms  are  of  the  second  order.     (Page  276.) 

Conjugate  surds  are  two  binomial  quadratic  surds  that  differ  only 
in  the  sign  of  one  of  the  terms. 

An  irrational,  or  radical,  equation  is  an  equation  containing  an 
irrational  root  of  the  unknown  number.     (Page  278.) 

CHAPTER  XXIII 

A  quadratic  equation  is  an  equation  of  the  second  degree  in  the 
unknown  number.     (Page  282.) 

The  constant  term  in  a  quadratic  equation  is  the  term  that  does 
not  contain  the  unknown  number. 

A  pure  quadratic  equation  is  an  equation  that  does  not  contain  the 
first  power  of  the  unknown  number. 

An  affected  quadratic  equation  is  an  equation  that  contains  both 
the  first  and  second  powers  of  the  unknown  number. 

Pure  quadratics  are  often  called  incomplete  quadratics,  and  affected 
quadratics  are  also  often  called  complete  quadratics. 

The  discriminant  of  the  roots  oi  ax"^-\-hx+c  =  0  is  b^  —  iac.  (Page 
298.)  

A  complex  number  is  a  number  of  the  form  a+6  v  —  1,  a  and  b 
denoting  real  numbers. 

Conjugate  complex  numbers  are  complex  numbers  which  differ 
in  the  sign  of  the  imaginary  term. 

CHAPTER  XXIV 

A  quadratic  equation  in  two  variables  is  an  equation  in  two  variables, 
one  or  both  of  which  are  of  the  second  degree.     (Page  305.) 

A  system  of  quadratic  equations  is  two  or  more  quadratic  equations 
considered  together. 

A  simultaneous  system  is  a  system  in  which  all  the  equations  can 
be  satisfied  by  the  same  values  of  the  variables. 


INDEX 


PAGE 

Absolute  value  of  a  number  .  22 

Addends 27 

Adding  indicated  products    .  27 
several  positive  and  nega- 
tive terms 29 

similar  terms 27-28 

Addition  and  subtraction  of 

fractions 188 

of  surds 270 

Addition  defined 27 

fundamental  laws  of  .    .    .  29 

law  of  grouping  for     ...  29 

law  of  order  for 29 

of  dissimilar  terms  ....  30 

of  monomials 27 

of  polynomials 32 

of  terms  partly  similar  .    .  48 

proportion  by 237 

analysis  of 238 

Affected  quadratic  equation  282 

solved  by  factoring     .    .    .  285 

Aggregation,  symbols  of     .   41-43 

Algebra  defined 7 

reasons  for  studying    .    .    .  1-6 

Algebraic  expression   ....  24 

value  of  an 26 

fraction 179 

function  defined 51 

functions 50 

language 8 

notation 24 

numbers 21,  50 

signs 9 

Alternation,  proportion  by     .  237 

Antecedent 229 

Approximate  values  of  surds  278 

Arranged  polynomials    ...  97 


PAGE 

Associative  law  of  addition   .  29 

of  multiplication      ....  94 

Assumption  for  irrational 

equations      280 

Axiom,  power       278 

root 287 

Axioms      13 

Balance  of  values 12 

Base  of  a  power       244 

Binomial  defined 25 

quadratic  surd 276 

surd 275 

theorem 248 

Binomials,  powers  of  ...    .  247 

Brace 42 

Bracket 42 

Check  or  test 14 

Check  on  algebraic  work 

defined 33 

Checking 16 

addition  by  substitution    .  33 

a  problem 101 

or  verifying  a  root      .    .    .  100 

Clearing  equations  of 

fractions 66 

principle  of 67 

application  of       103 

Clock  problems 117 

Coefficient 25 

of  a  radical 265 

Common  compound  factors  .  135 

Common  divisor      172 

fraction,  square  root  of  .    .  262 

multiple 175 


331 


332 


INDEX 


PAGE 

Comparison,  elimination  by  .    213 

Complete  divisor 256 

quadratic  equation      .    .    .   287 
approximate  values  of 

roots      .292 

normal  form 287 

roots  of  the 292 

quadratics 282 

Completing  the  square  .    .    .   288 

a  =  l       288 

a  not  1 289 

Complex  number 298 

Composition,  proportion  by  .   238 
Compound  expressions,  oper- 
ations on       43 

Conditional  equation      ...     61 

Conjugate  surds 276 

Conjugate  complex  numbers    298 

Consequent 229 

Constant 241 

term  of  a  quadratic     .    .    .   282 

Cube  defined 95 

root 140 

Definition  of  a*^    .    ..    .  108,  263 

of  a^ 263 

of  a-" 264 

Definitions,  summary  of   .    .  322 

Degree  of  an  equation    .    .    .  100 
Denominator  defined  .    .    .    .179 

Dependence  of  a  function  .    .  52 

Dependent  equations      ...  83 

Deriving  formulas 125 

Determinate  equations       .    .  85 

Difference  defined 35 

of  same  odd  powers    ...  153 

of  two  squares 143 

Digits,  Arabic 8 

Directed  numbers 21 

Directions  for  solving  equa- 
tions    101 


PAGE 

Discriminant  of  roots     .    .    .  298 

Dissimilar  terms      .....  25 

Distributive  law      95 

Dividend  defined 107 

Dividing  a  monomial  by  a 

monomial 107 

a  polynomial  by  a  mono- 
mial    109 

a  polynomial  by  a  poly- 
nomial    110 

Division  defined 107 

indicated 9 

of  fractions 193 

proportion  by 239 

sign  law  of 107 

Divisor,  common 172 

complete 256 

defined 107 

partial 256 

Double  meaning  of  +  and  —     23 

EHmination,  defined    ....  86 

by  addition  or  subtraction  87 

by  comparison 213 

by  substitution 120 

Ellipse 310 

Equation  defined 11 

degree  of 100 

determinate      85 

history  of      59,  60 

indeterminate 85 

linear 100 

literal  and  fractional  ...  198 

quadratic      282 

simple  or  linear 100 

in  quadratic  form    ....  293 

members  of  an 11 

root  of 61 

solving  an 13 

Equations,  dependent    .    .    .83 

inconsistent      83 


INDEX 


333 


PAGE 

Equations,  graphing  .    .    .   77-81 

linear 81 

non-simultaneous    ....  83 

simple 100 

simultaneous 82 

Equality,  sign  of     • 11 

Even  powers 95 

roots 252 

Evolution 250 

principle  of 274 

Examples  of  type-forms     .    .  131 

Expansions 247 

Exponent  in  multiplication    .  92 

in  product 93 

in  quotient 108 

law  of,  for  division      .    .    .  108 

law  of,  for  multiplication   .  93 

of  the  power 244 

zero,  meaning  of 108,  263 

Exponents,  fractional     .    .    .  272 

fundamental  laws  of   .    .    .  263 

theory  of       263 

Expression,  algebraic  24 

Extended  meaning  of  term    .  45 

Extremes  of  a  proportion  .    .  232 

Factor,  common 172 

defined 8 

highest  common 172 

rationalizing 275 

Factoring      134 

by  principles  of  quadratics  301 

Factors,  common  compound  135 

defined 8,  134 

monomial 134 

First  member 11 

Forming  quadratics  with 

given  roots 300 

Formula  defined 124 

Formulas  derived 125 

solved .  125 


PAGE 

Formulas,  solution  of      ...  210 

Fourth  proportional    ....  234 
Fraction  in  lowest  terms    .    .182 

improper 184 

proper 184 

Fractions,  addition  and  sub- 
traction of 188 

division  of 193 

multiphcation  of      ....  191 

Fractional  exponents  ....  272 

Function  defined     51 

dependence  of      52 

Function  of  a:,  n,  etc.      .    .    .  50 

fix),  fin),  etc 50 

Generalization  in  algebra  .    .   207 
General  number  defined     .    .123 

General  numbers 123 

quadratic      288 

quadratic  trinomial     ...    148 
Graphical  solution  of  one- 
letter  quadratics 

.    .    .     159-60,  282,  295 
of  quadratic  systems  .   305-313 

Graphing  data 74 

functions       50,  54 

equations      77-81 

Graph  of  a;2-a 283 

Higher  degree  equations  by 

factoring 286 

Highest  common  factor      .    .    172 

of  monomials 172 

of  polynomials 173 

Hyperbola 311,312 

Identity  defined 60 

sign  of 60 

Imaginary  number      ....  253 

roots      253 

Improper  fraction 184 


334 


INDEX 


PAGE 

Incomplete  quadratic 

equations 282 

trinomial  squares  ....  151 
Inconsistent  equations  ...  83 
Independent  equations  .  .  84,  86 
Independent  number  ....  50 
Indeterminate  equation     .    .     85 

Index  of  the  root 250 

Indicating  division      ....      -9 

multiplication  ....;.       9 

Inequality,  ratio  of  greater    .   231 

ratio  of  less 231 

signs  of 272 

Inversion,  proportion  by    .    .   237 

Involution .244 

Irrational  equations,  assump- 
tion for      280 

equations  in  one  unknown  -  278 
number 265 

Language,  using  algebraic  .  8 
Letters  representing  numbers  17 
Law  of  exponents  for  division  108 

for  multiplication  ....  93 
Law  of  grouping  for  addition     29 

for  multiplication  ....  94 
Law  of  order  for  addition  .    .     29 

for  multiplication    ....     94 

Linear  equations 81,  100 

Literal  and  fractional 

equations      198 

Lowest  common  denominator  187 
Lowest  common  multiple  .    .175 

of  monomials 175 


of  polynomials 


Meaning  of  exponent  0  .    108,  263 

type-forms 130 

Mean  proportional  ....  234 
Means  of  a  proportion  .  .  .  232 
Measuring  is  ratioing  .  .  .  230 
Members,  first  and  second     .     11 


176 


PAGE 

Mixed  number 184 

surd 265 

Mixed  surd  to  an  entire 

surd 270 

Monomial  defined 25 

Multiple,  common 175 

lowest  common 175 

Multiplicand  defined  ....  91 

Multiplication  defined    ...  91 

indicated       9 

law  of  exponents  for   .    .    .  93 

of  fractions 191 

of  surds 272 

sign  law  of 92 

Multiplier  defined 91 

negative 91 

Multiplying  monomials      .    .  93 
a  polynomial  by  a  mono- 
mial    96 

a  polynomial  by  a  poly- 
nomial         97 

Nature  of  roots  of  quadratic  298 

Negative  multiplier     ....  91 

Non-simultaneous  equations  83 

Notation 7 

algebraic 24 

in  problem-solving  .    .    .  15,  16 

system  of 24 

Number 13 

imaginary 253 

independent 50 

irrational 264 

mixed 184 

of  roots 252 

rational 264 

real- 253 

unknown 13 

Numbers,  directed 21 

general 123 

of  arithmetic 20 


INDEX 


335 


PAGE 

Numbers,  positive  and  nega- 
tive     21 

represented       8 

Numerator  defined      ....    179 

Odd  powers 95 

root 252 

Operations  on  compound  ex- 
pressions     43 

Opposite  qualities  of  alge- 
braic numbers      ....  21 

Order  of  a  radical 264 

second  and  third      ....  264 

Parabola 54,  306 

Parenthesis 41 

defined 42 

Partial  divisor      256 

Partly  similar  terms    ....  25 

Pascal's  triangle 248 

Picturing  functions     ....  54 

Polynomial,  arranged     ...  97 

defined 25 

square  root  of  a 254 

Polynomials  factored  by 

grouping 137 

Positive  and  negative  num- 
bers      21 

problems  in 23 

Power  defined 95 

base  of  the 244 

of  a  fraction 246 

of  a  monomial  surd     .    .    .  277 

of  a  product 246 

axiom 278 

second 95 

third      95 

Powers  and  roots 244 

of  binomials 247 

Primes  and  subscripts    .    .    .211 

Principal  root 254 


PAGE 

Principle  of  evolution     .    .    .  274 

Principles  of  proportion     .    .  235 

Problem,  general 207 

solving  a 15 

quadratics 302 

Problems  in  sumultaneous 

equations 106 

three  or  more  unknowns    .  226 

two  unknowns      ......  89 

Problem-solving,  suggestions 

on 113 

Product  defined 91 

of  sum  and  difference  of 

two  numbers 142 

of  two  binomials  with  a 

common  term 147 

of  two  numbers  equal  to  o  158 

Product,  sign  of  the    ....  91 

Products,  how  written    ...  8 

Proper  fraction 184 

Proportion  defined      ....  232 

by  addition 237 

by  addition  and  sub- 
traction       239 

by  alternation      237 

by  composition 238 

by  division 239 

by  inversion 237 

by  subtraction 238 

extremes  and  means  of  .    .  232 

principles  of 235 

Proportional,  mean     ....  234 

fourth 234 

third 234 

Proportionality,  test  of  .    .    .  233 

Pure  quadratic  equation    .    .  282 

normal  form  of 282 

solved  by  factoring     .    .    .  284 

Quadratic  equation,  affected  .  282 

nature  of  roots  of    ...    .  298 


336 


INDEX 


PAGE 

Quadratics,  pure 282 

Quadratic  equations    ....   282 
solved  by  formula   .    .    .    .291 

Quadratic  surd 265 

binomial 276 

trinomial       132 

Quality  of  number 21 

Quotient  defined      107 

Radical,  coefficient  of     .    .    .   265 

defined 250,  264 

degree  or  order  of    ...    .   264 

reduction  of 267 

sign !    ...   250 

Radicand      264 

Ratio,  antecedant  of       ...   229 

consequent  of       229 

defined 229 

of  greater  and  less  inequali- 
ty   231 

Rational  number 264 

Real  number 253 

roots      253 

Reasons  for  studying  algebra    1-6 

Reciprocal  of  a  number      .    .193 

Reduction  of  fractions    .    .    .    181 

of  improper  fractions      .    .    184 

of  mixed  expressions  .    .    .    186 

of  radicals 267 

of  surds  to  same  order    .    .271 
Remainder  in  subtraction      .     35 
Removing  symbols  of  aggre- 
gation     45 

Review  of  factoring    .    .    .    .156 

Root  of  a  fraction 252 

an  equation      61 

a  number      139,  250 

a  power 251 

a  product      251 

cube  and  square 140 

index  of  the      250 


PAGE 

Root,  principal 254 

square,  of  a  decimal  .  .  .261 
square,  of  numbers     .    .    .   259 

Roots,  imaginary 253 

of  complete  quadratic  .  .  292 
sets  of 86 

Satisfying  an  equation    ...  60 

Second  number 11 

power 95 

Sets  of  roots 86 

Signed  numbers 21 

Sign  law  of  division    ....  107 

of  multiplication      ....  92 

Sign  of  a  fraction 180 

continuation     .    .    .    .93,  245 

negative  numbers  ....  22 
positive  numbers     .    .    .    .22 

product 91 

quotient 107 

real  root 253 

Signs  of  inequality      ....  272 

Similar  terms 25 

Similar  with  respect  to  a  fac- 
tor        25 

Simultaneous  equations  de- 
fined    82 

system  of      82 

Simultaneous  simple  equations  85 
Solution  of  equations  by  fac- 
toring      15? 

formulas 210 

Solving  an  equation    ....  13 

a  problem 15 

equations  linear  in  -  and  -  215 

problems 88 

the  equation 15 

Solving  one-letter   equations 

graphically 55 


INDEX 


337 


PAGE 

Solving  formulas 125 

quadratics  by  factoring  284-285 
simultaneous  equations 

graphically 82-84 

Special  methods  for  systems 

of  quadratics    .    .    .    316-317 

Special  products      99 

Special  quadratic  trinomials  .    148 

Square  defined 95 

of  difference  of  two  numbers  138 
of  sum  of  two  numbers  .    .137 

Square  root 140 

of  a  binomial  surd       .    .    .    277 
of  a  common  fraction     .    .   262 

of  a  decimal 261 

of  a  polynomial    .....    254 

of  numbers 259 

Statement  in  problem-solv- 
ing         15-16 

Stating  and  formulating  laws  127 
Subscripts,  primes  and  .  .  .  211 
Substitution  defined    ....      60 

elimination  by 120 

Subtraction  defined     ....     35 

of  monomials 35 

of  polynomials 39 

proportion  by 238 

Subtrahend  defined     ....     35 
Subtracting,  defined  for  alge- 
bra      39 

dissimilar  terms 37 

monomials 38 

polynomials 39 

similar  terms 36 

terms  partly  similar    ...      49 
Suggestions  on  problem-solv- 
ing      113 

Summary  of  factoring    ...    155 

work  on  graphing    ....      58 

Sum  of  the  same  odd  powers    154 

Surd 265 


PAGE 

Surd,  pure  or  entire    ....  265 

mixed 265 

quadratic      265 

Surds,  addition  and  sub- 
traction of 270 

binomial        275 

binomial  quadratic      .    .    .  276 

conjugate      276 

division  of 274 

multipUcation  of      ....  272 
of  different  orders    .    .    .    .271 

similar 271 

Symbols  of  aggregation      .   41-43 

removing 45 

System  of  equations   ....  86 

System  of  notation      ....  24 

of  quadratic  equations    .    .  305 

Systems  solved  by  quadratics  305 

Term,  constant 282 

defined 24 

extended  meaning  of  .    .    .  45 

Terms,  dissimilar  and  similar  25 

of  a  fraction 179 

of  a  ratio 229 

partly  similar 25 

Test  of  proportionality  .    .    .  233 

of  roots  of  quadratic   .   ..    .  289 

Theorem,  binomial      ....  248 

Third  power 95 

proportional 234 

Three  or  more  unknowns  .    .  226 

To  check       16 

Transposition  applied     ...  63 

defined 62 

Triangle,  Pascal's 248 

Trinomial,  defined 25 

general  quadratic    .    .    .    .149 

quadratic 132 

squares      140 

squares,  incomplete     .        .  151 


338 


INDEX 


PAGE 

Type-forms,  meaning  of    .    .    130 

Type-forms,  interpreted     .    .132 

examples  of 131 

Unknown  number 13 

Value  of  an  algebraic  ex- 
pression       26 

of  any  letter 12 

of  a  quadratic  surd     .    ,    .  265 

of  a  ratio 229 


PAGE 

Values,  approximate,  of  roots 

of  quadratics 292 

Values  of  surds 210 

Variables 241 

Variation 241 

direct 241 

sign  of 241 

Varies  as,  or  directly  as     .    .  241 

Vinculum      42,  250 

Zero-exponent,  meaning 

of 108,  263 


VB  35929 


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